MATH 117 - ELEMENTS OF
MATHEMATICS - Jacobian matrix
Question Bank - Set 10
Liberty University
Question 1
Question
Let f:R3→R3be defined by f(x, y, z) = (x2y, yz, xz). Find the Jacobian
matrix of f.
Solution
To find the Jacobian matrix of f, we need to compute the partial derivatives of
each component function with respect to each input variable.
Step 1: Compute ∂f1
∂x ,∂f1
∂y , and ∂f1
∂z :
∂f1
∂x = 2xy, ∂f1
∂y =x2,∂f1
∂z = 0
Step 2: Compute ∂f2
∂x ,∂f2
∂y , and ∂f2
∂z :
∂f2
∂x = 0,∂f2
∂y =z, ∂f2
∂z =y
Step 3: Compute ∂f3
∂x ,∂f3
∂y , and ∂f3
∂z :
∂f3
∂x =z, ∂f3
∂y = 0,∂f3
∂z =x
Step 4: Assemble the partial derivatives into the Jacobian matrix:
Jf=
2xy x20
0z y
z0x
Therefore, the Jacobian matrix of fis given by Jf=
2xy x20
0z y
z0x
.
Question 2
Question
Let f(x, y, z) = (x2+y2, xy −z, z2) be a vector-valued function. Find the
Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to calculate the partial derivatives of
each component function.
Step 1: Calculate ∂f1
∂x ,∂f1
∂y , and ∂f1
∂z .
∂f1
∂x = 2x, ∂f1
∂y = 2y, ∂f1
∂z = 0
Step 2: Calculate ∂f2
∂x ,∂f2
∂y , and ∂f2
∂z .
∂f2
∂x =y, ∂f2
∂y =x, ∂f2
∂z =−1
Step 3: Calculate ∂f3
∂x ,∂f3
∂y , and ∂f3
∂z .
∂f3
∂x = 0,∂f3
∂y = 0,∂f3
∂z = 2z
Step 4: Assemble the Jacobian matrix Jf.
Jf=
2x2y0
y x −1
0 0 2z
Question 3
Question
Let f:R3→R2be a function defined by f(x, y, z) = (x2+yz, xyz). Determine
the Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to compute the partial derivatives of
each component function of fwith respect to the input variables.
Step 1: Find the partial derivatives of the first component function
2
Let g1(x, y, z) = x2+yz. We find the partial derivatives of g1as follows:
∂g1
∂x = 2x,
∂g1
∂y =z,
∂g1
∂z =y.
Step 2: Find the partial derivatives of the second component func-
tion
Let g2(x, y, z) = xyz. We find the partial derivatives of g2as follows:
∂g2
∂x =yz,
∂g2
∂y =xz,
∂g2
∂z =xy.
Step 3: Assemble the Jacobian matrix
The Jacobian matrix of fis the matrix of partial derivatives of the compo-
nent functions of f:
Jf= ∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z !=2x z y
yz xz xy.
Therefore, the Jacobian matrix of fis 2x z y
yz xz xy.
Question 4
Question
Let f:R3→R2be a function defined by f(x, y, z) = x2+yz, x
z. Determine
the Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to calculate the partial derivatives of
each component function.
Step 1: Find ∂f1
∂x ,∂f1
∂y , and ∂f1
∂z for the first component function f1(x, y, z) =
3
x2+yz.
∂f1
∂x = 2x,
∂f1
∂y =z,
∂f1
∂z =y.
Step 2: Find ∂f2
∂x ,∂f2
∂y , and ∂f2
∂z for the second component function f2(x, y, z) =
x
z.
∂f2
∂x =1
z,
∂f2
∂y = 0,
∂f2
∂z =−x
z2.
Step 3: Assemble the Jacobian matrix of fusing the partial derivatives
found in Step 1 and Step 2.
Jf="2x z y
1
z0−x
z2#.
Question 5
Question
Let f:R3→R2be a function defined by f(x, y, z) = (2x+y2, yz). Compute
the Jacobian matrix of f.
Solution
To compute the Jacobian matrix of f, we need to find the partial derivatives of
each component function with respect to each variable.
Step 1: Compute ∂f1∂x,∂f1
∂y , and ∂f1
∂z .
∂f1
∂x = 2,∂f1
∂y = 2y, ∂f1
∂z = 0
Step 2: Compute ∂f2∂x,∂f2
∂y , and ∂f2
∂z .
∂f2
∂x = 0,∂f2
∂y =z, ∂f2
∂z =y
4
Step 3: Assemble the Jacobian matrix. The Jacobian matrix of fis
given by
Jf="∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #
Substitute the partial derivatives we found in Steps 1 and 2 to get
Jf=2 2y0
0z y
Therefore, the Jacobian matrix of fis
Jf=2 2y0
0z y
Question 6
Question
Let f:R3→R2be a function given by:
f(x, y, z) = ex2+y2, z2
Compute the Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to compute the partial derivatives
of each component function with respect to the variables x,y, and z, and then
arrange them in a matrix.
Step 1: Find ∂f1
∂x ,∂f1
∂y , and ∂f1
∂z . Starting with the first component function:
f1(x, y, z) = ex2+y2
We have: ∂f1
∂x = 2xex2+y2
∂f1
∂y = 2yex2+y2
∂f1
∂z = 0
Step 2: Find ∂f2
∂x ,∂f2
∂y , and ∂f2
∂z . Moving on to the second component
function:
f2(x, y, z) = z2
We have: ∂f2
∂x = 0
5
∂f2
∂y = 0
∂f2
∂z = 2z
Step 3: Assemble the Jacobian matrix. The Jacobian matrix of fis given
by:
Jacobian(f) = "∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #
Substituting the partial derivatives we found:
Jacobian(f) = 2xex2+y22yex2+y20
0 0 2z
Question 7
Question
Let f:R3→R2be defined by f(x, y, z)=(x2+yz, y2−xz). Find the Jacobian
matrix of fat the point (1,2,−1).
Solution
Step 1: Calculate the partial derivatives of fwith respect to x,y, and z.
∂f1
∂x =∂
∂x (x2+yz)=2x
∂f1
∂y =∂
∂y (x2+yz) = z
∂f1
∂z =∂
∂z (x2+yz) = y
∂f2
∂x =∂
∂x (y2−xz) = −z
∂f2
∂y =∂
∂y (y2−xz)=2y
∂f2
∂z =∂
∂z (y2−xz) = −x
6
Step 2: Evaluate the partial derivatives at the point (1,2,−1).
∂f1
∂x (1,2,−1) = 2(1) = 2
∂f1
∂y (1,2,−1) = (−1) = −1
∂f1
∂z (1,2,−1) = 2
∂f2
∂x (1,2,−1) = 1
∂f2
∂y (1,2,−1) = 4
∂f2
∂z (1,2,−1) = −1
Step 3: Assemble the Jacobian matrix.
Jf(1,2,−1) = "∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #=2−1 2
1 4 −1
Question 8
Question
Let f(x, y, z)=(x2+y, 2y−z, 3z+x) be a vector-valued function in R3. Find
the Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to compute the partial derivatives of
the components of fwith respect to x,y, and z.
Step 1: Partial derivative of f1with respect to x:
∂f1
∂x =∂(x2+y)
∂x = 2x
Step 2: Partial derivative of f1with respect to y:
∂f1
∂y =∂(x2+y)
∂y = 1
Step 3: Partial derivative of f1with respect to z:
∂f1
∂z =∂(x2+y)
∂z = 0
Step 4: Partial derivative of f2with respect to x:
∂f2
∂x =∂(2y−z)
∂x = 0
7
Step 5: Partial derivative of f2with respect to y:
∂f2
∂y =∂(2y−z)
∂y = 2
Step 6: Partial derivative of f2with respect to z:
∂f2
∂z =∂(2y−z)
∂z =−1
Step 7: Partial derivative of f3with respect to x:
∂f3
∂x =∂(3z+x)
∂x = 1
Step 8: Partial derivative of f3with respect to y:
∂f3
∂y =∂(3z+x)
∂y = 0
Step 9: Partial derivative of f3with respect to z:
∂f3
∂z =∂(3z+x)
∂z = 3
Therefore, the Jacobian matrix of fis:
J(f) =
2 1 0
0 2 −1
1 0 3
Question 9
Question
Find the Jacobian matrix of the following function:
f(x, y, z)=(xy, x2z, y2z)
Solution
To find the Jacobian matrix of the function f(x, y, z) = (xy, x2z, y2z), we need
to compute the partial derivatives of each component function with respect to
each variable.
Step 1: Calculate the partial derivatives of fwith respect to x:
∂f1
∂x =∂
∂x (xy) = y
∂f2
∂x =∂
∂x (x2z)=2xz
∂f3
∂x =∂
∂x (y2z)=0
8
Step 2: Calculate the partial derivatives of fwith respect to y:
∂f1
∂y =∂
∂y (xy) = x
∂f2
∂y =∂
∂y (x2z)=0
∂f3
∂y =∂
∂y (y2z)=2yz
Step 3: Calculate the partial derivatives of fwith respect to z:
∂f1
∂z =∂
∂z (xy) = 0
∂f2
∂z =∂
∂z (x2z) = x2
∂f3
∂z =∂
∂z (y2z) = y2
Therefore, the Jacobian matrix of f(x, y, z)=(xy, x2z, y2z) is:
J=
y x 0
2xz 0 2yz
0x2y2
Question 10
Question
Let f:R3→R3be a mapping defined by f(x, y, z)=(xy, xz, yz). Determine
the Jacobian matrix of f.
Solution
To determine the Jacobian matrix of f, we need to compute the partial deriva-
tives of each component of fwith respect to each input variable.
Step 1: Calculate the partial derivatives of f:
∂f1
∂x =∂(xy)
∂x =y, ∂f1
∂y =∂(xy)
∂y =x, ∂f1
∂z = 0
∂f2
∂x =∂(xz)
∂x =z, ∂f2
∂y = 0,∂f2
∂z =∂(xz)
∂z =x
∂f3
∂x = 0,∂f3
∂y =∂(yz)
∂y =z, ∂f3
∂z =∂(yz)
∂z =y
Step 2: Assemble the Jacobian matrix: The Jacobian matrix of fis given
by
J(f) =
∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z
∂f3
∂x
∂f3
∂y
∂f3
∂z
=
y x 0
z0x
0z y
9
Therefore, the Jacobian matrix of fis
y x 0
z0x
0z y
.
Question 11
Question
Let f:R3→R2be a function defined by f(x, y, z)=(x2y, yz). Find the
Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to calculate the partial derivatives of
each component function with respect to each variable.
Step 1: Calculate ∂f1
∂x ,∂f1
∂y , and ∂f1
∂z for the first component function
f1(x, y, z) = x2y.
∂f1
∂x = 2xy
∂f1
∂y =x2
∂f1
∂z = 0
Step 2: Calculate ∂f2
∂x ,∂f2
∂y , and ∂f2
∂z for the second component function
f2(x, y, z) = yz.
∂f2
∂x = 0
∂f2
∂y =z
∂f2
∂z =y
Step 3: Assemble the partial derivatives into a matrix to form the Jacobian
matrix of f.
Jf=2xy x20
0z y
Question 12
Question
Let f:R3→R2be a function defined by f(x, y, z)=(xy2z, x2z). Find the
Jacobian matrix of fat the point (2,−1,3).
10
Solution
Step 1: Compute the partial derivatives of fwith respect to x, y, and z.
∂f
∂x =y2z, 2xz
∂f
∂y = (2xyz, 0)
∂f
∂z =xy2, x2
Step 2: Evaluate the partial derivatives at the point (2,−1,3).
∂f
∂x (2,−1,3) = (−1)2(3),2(2)(3)= (3,12)
∂f
∂y (2,−1,3) = (2(2)(−1)(3),0) = (−12,0)
∂f
∂z (2,−1,3) = 2(−1)2,22= (−2,4)
Step 3: Form the Jacobian matrix by arranging the partial derivatives as
rows.
Jf(2,−1,3) =
3 12
−12 0
−2 4
Question 13
Question
Let f:R3→R3be a vector field given by f(x, y, z)=(xy, xz, yz2). Find the
Jacobian matrix of f.
Solution
The Jacobian matrix of a vector field f:Rn→Rmis an m×nmatrix where
each entry is the partial derivative of the i-th component of fwith respect to
the j-th variable.
Step 1: Find the Jacobian matrix Jfof f. The components of fare
f1(x, y, z) = xy,f2(x, y, z) = xz, and f3(x, y, z) = yz2. We need to compute
the partial derivatives of each component with respect to each variable:
∂f1
∂x =y, ∂f1
∂y =x, ∂f1
∂z = 0,
∂f2
∂x =z, ∂f2
∂y = 0,∂f2
∂z =x,
∂f3
∂x = 0,∂f3
∂y =z2,∂f3
∂z = 2yz.
11
Therefore, the Jacobian matrix Jfis:
Jf=
∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z
∂f3
∂x
∂f3
∂y
∂f3
∂z
=
y x 0
z0x
0z22yz
.
Therefore, the Jacobian matrix of fis
y x 0
z0x
0z22yz
.
Question 14
Question
Let f(x, y, z)=(x2+y2, xy, xz) be a vector-valued function. Determine the
Jacobian matrix Jfof f.
Solution
To find the Jacobian matrix Jfof a vector-valued function f(x, y, z), we need
to compute the partial derivatives of each component function with respect to
each variable.
Step 1: Find the partial derivatives of fwith respect to x.
∂f
∂x =
∂
∂x (x2+y2)
∂
∂x (xy)
∂
∂x (xz)
=
2x
y
z
Step 2: Find the partial derivatives of fwith respect to y.
∂f
∂y =
∂
∂y (x2+y2)
∂
∂y (xy)
∂
∂y (xz)
=
2y
x
0
Step 3: Find the partial derivatives of fwith respect to z.
∂f
∂z =
∂
∂z (x2+y2)
∂
∂z (xy)
∂
∂z (xz)
=
0
0
x
Step 4: Assemble the partial derivatives into the Jacobian matrix Jf.
Jf=
2x2y0
y x 0
z0x
12
Question 15
Question
Let f(x, y, z) = x2+y2+z2and g(x, y, z) = exy . Find the Jacobian matrix of
the function g◦f, where g◦frepresents the composition of functions.
Solution
Step 1: Find the composition of functions g◦f(x, y, z).
g◦f(x, y, z) = g(f(x, y, z)) = e(x2+y2+z2)y
Step 2: Calculate the partial derivatives of g◦fwith respect to x, y, and z.
∂
∂x e(x2+y2+z2)y= 2xy2e(x2+y2+z2)y
∂
∂y e(x2+y2+z2)y= (x2+ 2y2+z2)e(x2+y2+z2)y
∂
∂z e(x2+y2+z2)y= 2yze(x2+y2+z2)y
Step 3: Assemble the Jacobian matrix of g◦f.
J(g◦f) = h2xy2e(x2+y2+z2)y(x2+ 2y2+z2)e(x2+y2+z2)y2yze(x2+y2+z2)yi
Therefore, the Jacobian matrix of g◦fis:
J(g◦f) = h2xy2e(x2+y2+z2)y(x2+ 2y2+z2)e(x2+y2+z2)y2yze(x2+y2+z2)yi
Question 16
Question
Let f:R3→R2be a differentiable function defined by f(x, y, z)=(x2y, yz).
Find the Jacobian matrix of f.
Solution
Step 1: The Jacobian matrix of fis given by the matrix of all first-order partial
derivatives of f, denoted by Jf.
Step 2: To find the Jacobian matrix of f, we need to compute the partial
derivatives of the components of fwith respect to x,y, and z.
Step 3: The components of fare f1(x, y, z) = x2yand f2(x, y, z) = yz.
13
Step 4: Let’s compute the partial derivatives of f1with respect to x,y, and
z:∂f1
∂x = 2xy,
∂f1
∂y =x2,
∂f1
∂z = 0.
Step 5: Now, let’s compute the partial derivatives of f2with respect to x,
y, and z:
∂f2
∂x = 0,
∂f2
∂y =z,
∂f2
∂z =y.
Step 6: The Jacobian matrix Jfis formed by these partial derivatives:
Jf=2xy x20
0z y.
Therefore, the Jacobian matrix of fis 2xy x20
0z y.
Question 17
Question
Let f:R3→R3be a vector field given by f(x, y, z)=(x2y, z, y2). Compute
the Jacobian matrix of f.
Solution
The Jacobian matrix of a vector field f:Rn→Rmis an m×nmatrix whose
entries are the partial derivatives of the components of f.
14
Step 1: Find the partial derivatives of the components of f.
∂
∂x (x2y)=2xy
∂
∂y (x2y) = x2
∂
∂z (x2y)=0
∂
∂x (z)=0
∂
∂y (z)=0
∂
∂z (z)=1
∂
∂x (y2)=0
∂
∂y (y2)=2y
∂
∂z (y2)=0
Step 2: Assemble the partial derivatives into the Jacobian matrix.
Jf=
2xy x20
0 0 1
0 2y0
Question 18
Question
Find the Jacobian matrix of the following function:
f(x, y, z) =
x2y
yz
x+z
Solution
To find the Jacobian matrix of a vector-valued function, we need to compute
the partial derivatives of each component function with respect to each variable.
Step 1: Compute the partial derivatives of fwith respect to x,y, and z.
Let’s start by finding the partial derivative of the first component function
x2ywith respect to x:
∂(x2y)
∂x = 2xy
15
Next, find the partial derivative of x2ywith respect to y:
∂(x2y)
∂y =x2
Moving on to the second component function yz, find the partial derivative
with respect to y:
∂(yz)
∂y =z
Finally, find the partial derivative of yz with respect to z:
∂(yz)
∂z =y
For the last component function x+z, find the partial derivative with respect
to x:∂(x+z)
∂x = 1
Next, find the partial derivative of x+zwith respect to z:
∂(x+z)
∂z = 1
Step 2: Assemble the partial derivatives into the Jacobian matrix.
Putting all the partial derivatives together, the Jacobian matrix of f(x, y, z)
is:
Jf(x, y, z) =
2xy x20
0z y
1 0 1
Question 19
Question
Consider the following system of equations:
(x2+y2= 25
xy = 9
Find the Jacobian matrix of the system at the point where x= 3 and y= 3.
Solution
To find the Jacobian matrix of the system at the given point, we first need
to find the partial derivatives of the functions F(x, y) = x2+y2−25 and
G(x, y) = xy −9.
Step 1: Find partial derivatives
16
For F(x, y) = x2+y2−25:
∂F
∂x = 2x
∂F
∂y = 2y
For G(x, y) = xy −9:
∂G
∂x =y
∂G
∂y =x
Step 2: Evaluate partial derivatives at the given point At the point
where x= 3 and y= 3:
∂F
∂x = 2(3) = 6
∂F
∂y = 2(3) = 6
∂G
∂x = 3
∂G
∂y = 3
Step 3: Assemble the Jacobian matrix The Jacobian matrix is given
by:
J(x, y) = "∂F
∂x
∂F
∂y
∂G
∂x
∂G
∂y #
Substitute the evaluated partial derivatives at the point x= 3 and y= 3
into the Jacobian matrix:
J(3,3) = 6 6
3 3
Therefore, the Jacobian matrix of the system at the point where x= 3 and
y= 3 is:
6 6
3 3
Question 20
Question
Let f:R2→R2be a function defined by f(x, y) = (x2+y2, xy). Find the
Jacobian matrix of fat the point (1,2).
17
Solution
Step 1: Compute the partial derivatives of fwith respect to xand y.
∂f1
∂x =∂
∂x (x2+y2)=2x,
∂f1
∂y =∂
∂y (x2+y2)=2y,
∂f2
∂x =∂
∂x (xy) = y,
∂f2
∂y =∂
∂y (xy) = x.
Step 2: Evaluate the partial derivatives at the point (1,2).
∂f1
∂x (1,2)
= 2(1) = 2,
∂f1
∂y (1,2)
= 2(2) = 4,
∂f2
∂x (1,2)
= 2,
∂f2
∂y (1,2)
= 1.
Step 3: Assemble the Jacobian matrix at (1,2). The Jacobian matrix of f
at (1,2) is given by
Jf(1,2) = 2 4
2 1.
Question 21
Question
Let f:R3→R3be defined by f(x, y, z) = (x2+yz, yz −x2, x +y+z). Find
the Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to compute the partial derivatives of
each component function.
Step 1: Compute the partial derivatives of fwith respect to x.
∂f1
∂x = 2x, ∂f2
∂x =−2x, ∂f3
∂x = 1
18
Step 2: Compute the partial derivatives of fwith respect to y.
∂f1
∂y =z, ∂f2
∂y =z, ∂f3
∂y = 1
Step 3: Compute the partial derivatives of fwith respect to z.
∂f1
∂z =y, ∂f2
∂z =y, ∂f3
∂z = 1
Step 4: Assemble the partial derivatives into the Jacobian matrix:
Jf=
∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z
∂f3
∂x
∂f3
∂y
∂f3
∂z
=
2x z y
−2x z y
1 1 1
Question 22
Question
Let f:R3→R2be a function defined by f(x, y, z) = (2x+y, 3y−z). Determine
the Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to compute the partial derivatives of
each component function with respect to each variable.
Step 1: Compute the partial derivatives of f: Let f(x, y, z)=(u, v) where
u= 2x+yand v= 3y−z. The partial derivatives are:
∂u
∂x = 2,∂u
∂y = 1,∂u
∂z = 0
∂v
∂x = 0,∂v
∂y = 3,∂v
∂z =−1
Step 2: Assemble the Jacobian matrix: The Jacobian matrix of fis a 2 ×3
matrix where each row corresponds to a component function and each column
corresponds to a variable. In this case, the Jacobian matrix is:
"∂u
∂x
∂u
∂y
∂u
∂z
∂v
∂x
∂v
∂y
∂v
∂z #=2 1 0
0 3 −1
Therefore, the Jacobian matrix of fis 2 1 0
0 3 −1.
19
Question 23
Question
Let f:R3→R2be defined by f(x, y, z) = (xy, yz). Calculate the Jacobian
matrix of f.
Solution
To find the Jacobian matrix of f, we need to calculate the partial derivatives of
each component of fwith respect to each input variable.
Step 1: Find ∂f1
∂x ,∂f1
∂y , and ∂f1
∂z .
Given f(x, y, z) = (xy, yz), we have:
∂f1
∂x =y
∂f1
∂y =x
∂f1
∂z = 0
Step 2: Find ∂f2
∂x ,∂f2
∂y , and ∂f2
∂z .
Given f(x, y, z) = (xy, yz), we have:
∂f2
∂x = 0
∂f2
∂y =z
∂f2
∂z =y
Step 3: Assemble the Jacobian matrix.
The Jacobian matrix of fis given by:
Jf="∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #
Substitute the partial derivatives we found into the Jacobian matrix:
Jf=y x 0
0z y
Question 24
Question
Let f(x, y, z)=(x2+y2, xyz, z3). Find the Jacobian matrix of fat the point
(1,2,3).
20
Solution
Step 1: First, we need to calculate the partial derivatives of fwith respect to
each variable x,y, and z.
Step 2: The Jacobian matrix of fis given by
Jf=
∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z
∂f3
∂x
∂f3
∂y
∂f3
∂z
Step 3: Calculating the partial derivatives, we get
∂f1
∂x = 2x, ∂f1
∂y = 2y, ∂f1
∂z = 0
∂f2
∂x =yz, ∂f2
∂y =xz, ∂f2
∂z =xy
∂f3
∂x = 0,∂f3
∂y = 0,∂f3
∂z = 3z2
Step 4: Evaluating the partial derivatives at the point (1,2,3), we have
∂f1
∂x = 2(1) = 2,∂f1
∂y = 2(2) = 4,∂f1
∂z = 0
∂f2
∂x = 2,∂f2
∂y = 3,∂f2
∂z = 2
∂f3
∂x = 0,∂f3
∂y = 0,∂f3
∂z = 27
Step 5: Therefore, the Jacobian matrix of fat (1,2,3) is
Jf=
2 4 0
2 3 2
0 0 27
Question 25
Question
Let f:R3→R2and g:R2→R3be defined by
f(x, y, z) = (xy, xz)
g(u, v)=(u+v, uv, u −v)
Find the Jacobian matrix of the composition function (g◦f)(x, y, z) and
evaluate it at the point (1,2,3).
21
Solution
Step 1: Calculate the composition function (g◦f)(x, y, z).
(g◦f)(x, y, z) = g(f(x, y, z)) = g(xy, xz)=(xy +xz, xy ·xz, xy −xz)
Step 2: Find the Jacobian matrix of (g◦f)(x, y, z). To find the Jacobian
matrix, we will compute the partial derivatives of each component of (g◦f)
with respect to x,y, and z.
Let u=xy +xz,v=xyz, and w=xy −xz. Then, we have
∂u
∂x =y+z
∂u
∂y =x
∂u
∂z =x
∂v
∂x =yz
∂v
∂y =xz
∂v
∂z =xy
∂w
∂x =y−z
∂w
∂y =x
∂w
∂z =−x
Therefore, the Jacobian matrix of (g◦f)(x, y, z) is
J=
y+z x x
yz xz xy
y−z x −x
Step 3: Evaluate the Jacobian matrix at the point (1, 2, 3). Substitute
x= 1, y= 2, and z= 3 into the Jacobian matrix Jto get
J(1,2,3) =
511
662
−1 1 −1
22
Question 26
Question
Find the Jacobian matrix Jof the transformation given by the equations
x=usin(v), y =ucos(v)
Solution
Step 1: Compute the partial derivatives of xand ywith respect to uand v.
∂x
∂u = sin(v),∂x
∂v =ucos(v),
∂y
∂u = cos(v),∂y
∂v =−usin(v).
Step 2: Assemble the Jacobian matrix using the partial derivatives.
J=∂x
∂u
∂x
∂v
∂y
∂u
∂y
∂v =sin(v)ucos(v)
cos(v)−usin(v)
Therefore, the Jacobian matrix Jof the transformation is:
J=sin(v)ucos(v)
cos(v)−usin(v)
Question 27
Question
Let f:R3→R2be a vector-valued function defined by f(x, y, z) = (x2+
yz, z2−xy). Find the Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to compute the partial derivatives of
each component function with respect to each variable.
Step 1: Find the partial derivatives of the first component f1(x, y, z) =
x2+yz.
∂f1
∂x = 2x,
∂f1
∂y =z,
∂f1
∂z =y.
23
Step 2: Find the partial derivatives of the second component f2(x, y, z) =
z2−xy.
∂f2
∂x =−y,
∂f2
∂y =−x,
∂f2
∂z = 2z.
Step 3: Assemble the partial derivatives into the Jacobian matrix Jf:
Jf="∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #=2x z y
−y−x2z.
Therefore, the Jacobian matrix of fis
Jf=2x z y
−y−x2z.
Question 28
Question
Let f(x, y) = (x2+y2, xy). Find the Jacobian matrix Jf(x, y) of the function
f.
Solution
To find the Jacobian matrix Jf(x, y) of the function f, we need to compute the
partial derivatives of fwith respect to each variable.
Step 1: Compute ∂f1
∂x and ∂f1
∂y :
∂f1
∂x =∂
∂x (x2+y2)=2x
∂f1
∂y =∂
∂y (x2+y2)=2y
Step 2: Compute ∂f2
∂x and ∂f2
∂y :
∂f2
∂x =∂
∂x (xy) = y
∂f2
∂y =∂
∂y (xy) = x
24
Step 3: Assemble the partial derivatives into the Jacobian matrix:
Jf(x, y) = "∂f1
∂x
∂f1
∂y
∂f2
∂x
∂f2
∂y #=2x2y
y x
Therefore, the Jacobian matrix of the function f(x, y) = (x2+y2, xy) is
2x2y
y x .
Question 29
Question
Let f(x, y, z)=(x2y+z, y2z, xyz) be a vector-valued function. Compute the
Jacobian matrix of f.
Solution
To compute the Jacobian matrix of f, we need to find the partial derivatives of
each component function with respect to each input variable.
Step 1: Find the partial derivatives of fwith respect to x.
The first component of fis f1(x, y, z) = x2y+z. Taking the partial derivative
of f1with respect to x, we get:
∂f1
∂x =∂
∂x (x2y+z)=2xy
Step 2: Find the partial derivatives of fwith respect to y.
The second component of fis f2(x, y, z) = y2z. Taking the partial derivative
of f2with respect to y, we have:
∂f2
∂y =∂
∂y (y2z)=2yz
Step 3: Find the partial derivatives of fwith respect to z.
The third component of fis f3(x, y, z) = xyz. Taking the partial derivative
of f3with respect to z, we obtain:
∂f3
∂z =∂
∂z (xyz) = xy
Step 4: Assemble the partial derivatives into the Jacobian matrix.
The Jacobian matrix of fwill be:
Jf(x, y, z) =
2xy x21
0 2yz 0
y x 0
Therefore, the Jacobian matrix of fis
2xy x21
0 2yz 0
y x 0
.
25
Question 30
Question
Let f:R3→R2be a function defined by f(x, y, z) = (x2y, z). Find the
Jacobian matrix of fat the point (1,1,1).
Solution
Step 1: Compute the partial derivatives of f(x, y, z).
∂f
∂x =∂
∂x (x2y)
∂
∂x (z)=2xy
0
∂f
∂y = ∂
∂y (x2y)
∂
∂y (z)!=x2
0
∂f
∂z =∂
∂z (x2y)
∂
∂z (z)=0
1
Step 2: Evaluate the partial derivatives at the point (1,1,1).
∂f
∂x (1,1,1) = 2·1·1
0=2
0
∂f
∂y (1,1,1) = 12
0=1
0
∂f
∂z (1,1,1) = 0
1
Step 3: Assemble the Jacobian matrix. The Jacobian matrix Jf(1,1,1) is
given by
Jf(1,1,1) = ∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z !=210
001
26
Question 2
Question
Let f(x, y, z) = (x2+y2, xy −z, z2) be a vector-valued function. Find the
Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to calculate the partial derivatives of
each component function.
Step 1: Calculate ∂f1
∂x ,∂f1
∂y , and ∂f1
∂z .
∂f1
∂x = 2x, ∂f1
∂y = 2y, ∂f1
∂z = 0
Step 2: Calculate ∂f2
∂x ,∂f2
∂y , and ∂f2
∂z .
∂f2
∂x =y, ∂f2
∂y =x, ∂f2
∂z =−1
Step 3: Calculate ∂f3
∂x ,∂f3
∂y , and ∂f3
∂z .
∂f3
∂x = 0,∂f3
∂y = 0,∂f3
∂z = 2z
Step 4: Assemble the Jacobian matrix Jf.
Jf=
2x2y0
y x −1
0 0 2z
Question 3
Question
Let f:R3→R2be a function defined by f(x, y, z) = (x2+yz, xyz). Determine
the Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to compute the partial derivatives of
each component function of fwith respect to the input variables.
Step 1: Find the partial derivatives of the first component function
2
Let g1(x, y, z) = x2+yz. We find the partial derivatives of g1as follows:
∂g1
∂x = 2x,
∂g1
∂y =z,
∂g1
∂z =y.
Step 2: Find the partial derivatives of the second component func-
tion
Let g2(x, y, z) = xyz. We find the partial derivatives of g2as follows:
∂g2
∂x =yz,
∂g2
∂y =xz,
∂g2
∂z =xy.
Step 3: Assemble the Jacobian matrix
The Jacobian matrix of fis the matrix of partial derivatives of the compo-
nent functions of f:
Jf= ∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z !=2x z y
yz xz xy.
Therefore, the Jacobian matrix of fis 2x z y
yz xz xy.
Question 4
Question
Let f:R3→R2be a function defined by f(x, y, z) = x2+yz, x
z. Determine
the Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to calculate the partial derivatives of
each component function.
Step 1: Find ∂f1
∂x ,∂f1
∂y , and ∂f1
∂z for the first component function f1(x, y, z) =
3
x2+yz.
∂f1
∂x = 2x,
∂f1
∂y =z,
∂f1
∂z =y.
Step 2: Find ∂f2
∂x ,∂f2
∂y , and ∂f2
∂z for the second component function f2(x, y, z) =
x
z.
∂f2
∂x =1
z,
∂f2
∂y = 0,
∂f2
∂z =−x
z2.
Step 3: Assemble the Jacobian matrix of fusing the partial derivatives
found in Step 1 and Step 2.
Jf="2x z y
1
z0−x
z2#.
Question 5
Question
Let f:R3→R2be a function defined by f(x, y, z) = (2x+y2, yz). Compute
the Jacobian matrix of f.
Solution
To compute the Jacobian matrix of f, we need to find the partial derivatives of
each component function with respect to each variable.
Step 1: Compute ∂f1∂x,∂f1
∂y , and ∂f1
∂z .
∂f1
∂x = 2,∂f1
∂y = 2y, ∂f1
∂z = 0
Step 2: Compute ∂f2∂x,∂f2
∂y , and ∂f2
∂z .
∂f2
∂x = 0,∂f2
∂y =z, ∂f2
∂z =y
4
Step 3: Assemble the Jacobian matrix. The Jacobian matrix of fis
given by
Jf="∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #
Substitute the partial derivatives we found in Steps 1 and 2 to get
Jf=2 2y0
0z y
Therefore, the Jacobian matrix of fis
Jf=2 2y0
0z y
Question 6
Question
Let f:R3→R2be a function given by:
f(x, y, z) = ex2+y2, z2
Compute the Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to compute the partial derivatives
of each component function with respect to the variables x,y, and z, and then
arrange them in a matrix.
Step 1: Find ∂f1
∂x ,∂f1
∂y , and ∂f1
∂z . Starting with the first component function:
f1(x, y, z) = ex2+y2
We have: ∂f1
∂x = 2xex2+y2
∂f1
∂y = 2yex2+y2
∂f1
∂z = 0
Step 2: Find ∂f2
∂x ,∂f2
∂y , and ∂f2
∂z . Moving on to the second component
function:
f2(x, y, z) = z2
We have: ∂f2
∂x = 0
5
∂f2
∂y = 0
∂f2
∂z = 2z
Step 3: Assemble the Jacobian matrix. The Jacobian matrix of fis given
by:
Jacobian(f) = "∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #
Substituting the partial derivatives we found:
Jacobian(f) = 2xex2+y22yex2+y20
0 0 2z
Question 7
Question
Let f:R3→R2be defined by f(x, y, z)=(x2+yz, y2−xz). Find the Jacobian
matrix of fat the point (1,2,−1).
Solution
Step 1: Calculate the partial derivatives of fwith respect to x,y, and z.
∂f1
∂x =∂
∂x (x2+yz)=2x
∂f1
∂y =∂
∂y (x2+yz) = z
∂f1
∂z =∂
∂z (x2+yz) = y
∂f2
∂x =∂
∂x (y2−xz) = −z
∂f2
∂y =∂
∂y (y2−xz)=2y
∂f2
∂z =∂
∂z (y2−xz) = −x
6
Step 2: Evaluate the partial derivatives at the point (1,2,−1).
∂f1
∂x (1,2,−1) = 2(1) = 2
∂f1
∂y (1,2,−1) = (−1) = −1
∂f1
∂z (1,2,−1) = 2
∂f2
∂x (1,2,−1) = 1
∂f2
∂y (1,2,−1) = 4
∂f2
∂z (1,2,−1) = −1
Step 3: Assemble the Jacobian matrix.
Jf(1,2,−1) = "∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #=2−1 2
1 4 −1
Question 8
Question
Let f(x, y, z)=(x2+y, 2y−z, 3z+x) be a vector-valued function in R3. Find
the Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to compute the partial derivatives of
the components of fwith respect to x,y, and z.
Step 1: Partial derivative of f1with respect to x:
∂f1
∂x =∂(x2+y)
∂x = 2x
Step 2: Partial derivative of f1with respect to y:
∂f1
∂y =∂(x2+y)
∂y = 1
Step 3: Partial derivative of f1with respect to z:
∂f1
∂z =∂(x2+y)
∂z = 0
Step 4: Partial derivative of f2with respect to x:
∂f2
∂x =∂(2y−z)
∂x = 0
7
Step 5: Partial derivative of f2with respect to y:
∂f2
∂y =∂(2y−z)
∂y = 2
Step 6: Partial derivative of f2with respect to z:
∂f2
∂z =∂(2y−z)
∂z =−1
Step 7: Partial derivative of f3with respect to x:
∂f3
∂x =∂(3z+x)
∂x = 1
Step 8: Partial derivative of f3with respect to y:
∂f3
∂y =∂(3z+x)
∂y = 0
Step 9: Partial derivative of f3with respect to z:
∂f3
∂z =∂(3z+x)
∂z = 3
Therefore, the Jacobian matrix of fis:
J(f) =
2 1 0
0 2 −1
1 0 3
Question 9
Question
Find the Jacobian matrix of the following function:
f(x, y, z)=(xy, x2z, y2z)
Solution
To find the Jacobian matrix of the function f(x, y, z) = (xy, x2z, y2z), we need
to compute the partial derivatives of each component function with respect to
each variable.
Step 1: Calculate the partial derivatives of fwith respect to x:
∂f1
∂x =∂
∂x (xy) = y
∂f2
∂x =∂
∂x (x2z)=2xz
∂f3
∂x =∂
∂x (y2z)=0
8
Step 2: Calculate the partial derivatives of fwith respect to y:
∂f1
∂y =∂
∂y (xy) = x
∂f2
∂y =∂
∂y (x2z)=0
∂f3
∂y =∂
∂y (y2z)=2yz
Step 3: Calculate the partial derivatives of fwith respect to z:
∂f1
∂z =∂
∂z (xy) = 0
∂f2
∂z =∂
∂z (x2z) = x2
∂f3
∂z =∂
∂z (y2z) = y2
Therefore, the Jacobian matrix of f(x, y, z)=(xy, x2z, y2z) is:
J=
y x 0
2xz 0 2yz
0x2y2
Question 10
Question
Let f:R3→R3be a mapping defined by f(x, y, z)=(xy, xz, yz). Determine
the Jacobian matrix of f.
Solution
To determine the Jacobian matrix of f, we need to compute the partial deriva-
tives of each component of fwith respect to each input variable.
Step 1: Calculate the partial derivatives of f:
∂f1
∂x =∂(xy)
∂x =y, ∂f1
∂y =∂(xy)
∂y =x, ∂f1
∂z = 0
∂f2
∂x =∂(xz)
∂x =z, ∂f2
∂y = 0,∂f2
∂z =∂(xz)
∂z =x
∂f3
∂x = 0,∂f3
∂y =∂(yz)
∂y =z, ∂f3
∂z =∂(yz)
∂z =y
Step 2: Assemble the Jacobian matrix: The Jacobian matrix of fis given
by
J(f) =
∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z
∂f3
∂x
∂f3
∂y
∂f3
∂z
=
y x 0
z0x
0z y
9
Therefore, the Jacobian matrix of fis
y x 0
z0x
0z y
.
Question 11
Question
Let f:R3→R2be a function defined by f(x, y, z)=(x2y, yz). Find the
Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to calculate the partial derivatives of
each component function with respect to each variable.
Step 1: Calculate ∂f1
∂x ,∂f1
∂y , and ∂f1
∂z for the first component function
f1(x, y, z) = x2y.
∂f1
∂x = 2xy
∂f1
∂y =x2
∂f1
∂z = 0
Step 2: Calculate ∂f2
∂x ,∂f2
∂y , and ∂f2
∂z for the second component function
f2(x, y, z) = yz.
∂f2
∂x = 0
∂f2
∂y =z
∂f2
∂z =y
Step 3: Assemble the partial derivatives into a matrix to form the Jacobian
matrix of f.
Jf=2xy x20
0z y
Question 12
Question
Let f:R3→R2be a function defined by f(x, y, z)=(xy2z, x2z). Find the
Jacobian matrix of fat the point (2,−1,3).
10
Solution
Step 1: Compute the partial derivatives of fwith respect to x, y, and z.
∂f
∂x =y2z, 2xz
∂f
∂y = (2xyz, 0)
∂f
∂z =xy2, x2
Step 2: Evaluate the partial derivatives at the point (2,−1,3).
∂f
∂x (2,−1,3) = (−1)2(3),2(2)(3)= (3,12)
∂f
∂y (2,−1,3) = (2(2)(−1)(3),0) = (−12,0)
∂f
∂z (2,−1,3) = 2(−1)2,22= (−2,4)
Step 3: Form the Jacobian matrix by arranging the partial derivatives as
rows.
Jf(2,−1,3) =
3 12
−12 0
−2 4
Question 13
Question
Let f:R3→R3be a vector field given by f(x, y, z)=(xy, xz, yz2). Find the
Jacobian matrix of f.
Solution
The Jacobian matrix of a vector field f:Rn→Rmis an m×nmatrix where
each entry is the partial derivative of the i-th component of fwith respect to
the j-th variable.
Step 1: Find the Jacobian matrix Jfof f. The components of fare
f1(x, y, z) = xy,f2(x, y, z) = xz, and f3(x, y, z) = yz2. We need to compute
the partial derivatives of each component with respect to each variable:
∂f1
∂x =y, ∂f1
∂y =x, ∂f1
∂z = 0,
∂f2
∂x =z, ∂f2
∂y = 0,∂f2
∂z =x,
∂f3
∂x = 0,∂f3
∂y =z2,∂f3
∂z = 2yz.
11
Therefore, the Jacobian matrix Jfis:
Jf=
∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z
∂f3
∂x
∂f3
∂y
∂f3
∂z
=
y x 0
z0x
0z22yz
.
Therefore, the Jacobian matrix of fis
y x 0
z0x
0z22yz
.
Question 14
Question
Let f(x, y, z)=(x2+y2, xy, xz) be a vector-valued function. Determine the
Jacobian matrix Jfof f.
Solution
To find the Jacobian matrix Jfof a vector-valued function f(x, y, z), we need
to compute the partial derivatives of each component function with respect to
each variable.
Step 1: Find the partial derivatives of fwith respect to x.
∂f
∂x =
∂
∂x (x2+y2)
∂
∂x (xy)
∂
∂x (xz)
=
2x
y
z
Step 2: Find the partial derivatives of fwith respect to y.
∂f
∂y =
∂
∂y (x2+y2)
∂
∂y (xy)
∂
∂y (xz)
=
2y
x
0
Step 3: Find the partial derivatives of fwith respect to z.
∂f
∂z =
∂
∂z (x2+y2)
∂
∂z (xy)
∂
∂z (xz)
=
0
0
x
Step 4: Assemble the partial derivatives into the Jacobian matrix Jf.
Jf=
2x2y0
y x 0
z0x
12
Question 15
Question
Let f(x, y, z) = x2+y2+z2and g(x, y, z) = exy . Find the Jacobian matrix of
the function g◦f, where g◦frepresents the composition of functions.
Solution
Step 1: Find the composition of functions g◦f(x, y, z).
g◦f(x, y, z) = g(f(x, y, z)) = e(x2+y2+z2)y
Step 2: Calculate the partial derivatives of g◦fwith respect to x, y, and z.
∂
∂x e(x2+y2+z2)y= 2xy2e(x2+y2+z2)y
∂
∂y e(x2+y2+z2)y= (x2+ 2y2+z2)e(x2+y2+z2)y
∂
∂z e(x2+y2+z2)y= 2yze(x2+y2+z2)y
Step 3: Assemble the Jacobian matrix of g◦f.
J(g◦f) = h2xy2e(x2+y2+z2)y(x2+ 2y2+z2)e(x2+y2+z2)y2yze(x2+y2+z2)yi
Therefore, the Jacobian matrix of g◦fis:
J(g◦f) = h2xy2e(x2+y2+z2)y(x2+ 2y2+z2)e(x2+y2+z2)y2yze(x2+y2+z2)yi
Question 16
Question
Let f:R3→R2be a differentiable function defined by f(x, y, z)=(x2y, yz).
Find the Jacobian matrix of f.
Solution
Step 1: The Jacobian matrix of fis given by the matrix of all first-order partial
derivatives of f, denoted by Jf.
Step 2: To find the Jacobian matrix of f, we need to compute the partial
derivatives of the components of fwith respect to x,y, and z.
Step 3: The components of fare f1(x, y, z) = x2yand f2(x, y, z) = yz.
13
Step 4: Let’s compute the partial derivatives of f1with respect to x,y, and
z:∂f1
∂x = 2xy,
∂f1
∂y =x2,
∂f1
∂z = 0.
Step 5: Now, let’s compute the partial derivatives of f2with respect to x,
y, and z:
∂f2
∂x = 0,
∂f2
∂y =z,
∂f2
∂z =y.
Step 6: The Jacobian matrix Jfis formed by these partial derivatives:
Jf=2xy x20
0z y.
Therefore, the Jacobian matrix of fis 2xy x20
0z y.
Question 17
Question
Let f:R3→R3be a vector field given by f(x, y, z)=(x2y, z, y2). Compute
the Jacobian matrix of f.
Solution
The Jacobian matrix of a vector field f:Rn→Rmis an m×nmatrix whose
entries are the partial derivatives of the components of f.
14
Step 1: Find the partial derivatives of the components of f.
∂
∂x (x2y)=2xy
∂
∂y (x2y) = x2
∂
∂z (x2y)=0
∂
∂x (z)=0
∂
∂y (z)=0
∂
∂z (z)=1
∂
∂x (y2)=0
∂
∂y (y2)=2y
∂
∂z (y2)=0
Step 2: Assemble the partial derivatives into the Jacobian matrix.
Jf=
2xy x20
0 0 1
0 2y0
Question 18
Question
Find the Jacobian matrix of the following function:
f(x, y, z) =
x2y
yz
x+z
Solution
To find the Jacobian matrix of a vector-valued function, we need to compute
the partial derivatives of each component function with respect to each variable.
Step 1: Compute the partial derivatives of fwith respect to x,y, and z.
Let’s start by finding the partial derivative of the first component function
x2ywith respect to x:
∂(x2y)
∂x = 2xy
15
Next, find the partial derivative of x2ywith respect to y:
∂(x2y)
∂y =x2
Moving on to the second component function yz, find the partial derivative
with respect to y:
∂(yz)
∂y =z
Finally, find the partial derivative of yz with respect to z:
∂(yz)
∂z =y
For the last component function x+z, find the partial derivative with respect
to x:∂(x+z)
∂x = 1
Next, find the partial derivative of x+zwith respect to z:
∂(x+z)
∂z = 1
Step 2: Assemble the partial derivatives into the Jacobian matrix.
Putting all the partial derivatives together, the Jacobian matrix of f(x, y, z)
is:
Jf(x, y, z) =
2xy x20
0z y
1 0 1
Question 19
Question
Consider the following system of equations:
(x2+y2= 25
xy = 9
Find the Jacobian matrix of the system at the point where x= 3 and y= 3.
Solution
To find the Jacobian matrix of the system at the given point, we first need
to find the partial derivatives of the functions F(x, y) = x2+y2−25 and
G(x, y) = xy −9.
Step 1: Find partial derivatives
16
For F(x, y) = x2+y2−25:
∂F
∂x = 2x
∂F
∂y = 2y
For G(x, y) = xy −9:
∂G
∂x =y
∂G
∂y =x
Step 2: Evaluate partial derivatives at the given point At the point
where x= 3 and y= 3:
∂F
∂x = 2(3) = 6
∂F
∂y = 2(3) = 6
∂G
∂x = 3
∂G
∂y = 3
Step 3: Assemble the Jacobian matrix The Jacobian matrix is given
by:
J(x, y) = "∂F
∂x
∂F
∂y
∂G
∂x
∂G
∂y #
Substitute the evaluated partial derivatives at the point x= 3 and y= 3
into the Jacobian matrix:
J(3,3) = 6 6
3 3
Therefore, the Jacobian matrix of the system at the point where x= 3 and
y= 3 is:
6 6
3 3
Question 20
Question
Let f:R2→R2be a function defined by f(x, y) = (x2+y2, xy). Find the
Jacobian matrix of fat the point (1,2).
17
Solution
Step 1: Compute the partial derivatives of fwith respect to xand y.
∂f1
∂x =∂
∂x (x2+y2)=2x,
∂f1
∂y =∂
∂y (x2+y2)=2y,
∂f2
∂x =∂
∂x (xy) = y,
∂f2
∂y =∂
∂y (xy) = x.
Step 2: Evaluate the partial derivatives at the point (1,2).
∂f1
∂x (1,2)
= 2(1) = 2,
∂f1
∂y (1,2)
= 2(2) = 4,
∂f2
∂x (1,2)
= 2,
∂f2
∂y (1,2)
= 1.
Step 3: Assemble the Jacobian matrix at (1,2). The Jacobian matrix of f
at (1,2) is given by
Jf(1,2) = 2 4
2 1.
Question 21
Question
Let f:R3→R3be defined by f(x, y, z) = (x2+yz, yz −x2, x +y+z). Find
the Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to compute the partial derivatives of
each component function.
Step 1: Compute the partial derivatives of fwith respect to x.
∂f1
∂x = 2x, ∂f2
∂x =−2x, ∂f3
∂x = 1
18
Step 2: Compute the partial derivatives of fwith respect to y.
∂f1
∂y =z, ∂f2
∂y =z, ∂f3
∂y = 1
Step 3: Compute the partial derivatives of fwith respect to z.
∂f1
∂z =y, ∂f2
∂z =y, ∂f3
∂z = 1
Step 4: Assemble the partial derivatives into the Jacobian matrix:
Jf=
∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z
∂f3
∂x
∂f3
∂y
∂f3
∂z
=
2x z y
−2x z y
1 1 1
Question 22
Question
Let f:R3→R2be a function defined by f(x, y, z) = (2x+y, 3y−z). Determine
the Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to compute the partial derivatives of
each component function with respect to each variable.
Step 1: Compute the partial derivatives of f: Let f(x, y, z)=(u, v) where
u= 2x+yand v= 3y−z. The partial derivatives are:
∂u
∂x = 2,∂u
∂y = 1,∂u
∂z = 0
∂v
∂x = 0,∂v
∂y = 3,∂v
∂z =−1
Step 2: Assemble the Jacobian matrix: The Jacobian matrix of fis a 2 ×3
matrix where each row corresponds to a component function and each column
corresponds to a variable. In this case, the Jacobian matrix is:
"∂u
∂x
∂u
∂y
∂u
∂z
∂v
∂x
∂v
∂y
∂v
∂z #=2 1 0
0 3 −1
Therefore, the Jacobian matrix of fis 2 1 0
0 3 −1.
19
Question 23
Question
Let f:R3→R2be defined by f(x, y, z) = (xy, yz). Calculate the Jacobian
matrix of f.
Solution
To find the Jacobian matrix of f, we need to calculate the partial derivatives of
each component of fwith respect to each input variable.
Step 1: Find ∂f1
∂x ,∂f1
∂y , and ∂f1
∂z .
Given f(x, y, z) = (xy, yz), we have:
∂f1
∂x =y
∂f1
∂y =x
∂f1
∂z = 0
Step 2: Find ∂f2
∂x ,∂f2
∂y , and ∂f2
∂z .
Given f(x, y, z) = (xy, yz), we have:
∂f2
∂x = 0
∂f2
∂y =z
∂f2
∂z =y
Step 3: Assemble the Jacobian matrix.
The Jacobian matrix of fis given by:
Jf="∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #
Substitute the partial derivatives we found into the Jacobian matrix:
Jf=y x 0
0z y
Question 24
Question
Let f(x, y, z)=(x2+y2, xyz, z3). Find the Jacobian matrix of fat the point
(1,2,3).
20
Solution
Step 1: First, we need to calculate the partial derivatives of fwith respect to
each variable x,y, and z.
Step 2: The Jacobian matrix of fis given by
Jf=
∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z
∂f3
∂x
∂f3
∂y
∂f3
∂z
Step 3: Calculating the partial derivatives, we get
∂f1
∂x = 2x, ∂f1
∂y = 2y, ∂f1
∂z = 0
∂f2
∂x =yz, ∂f2
∂y =xz, ∂f2
∂z =xy
∂f3
∂x = 0,∂f3
∂y = 0,∂f3
∂z = 3z2
Step 4: Evaluating the partial derivatives at the point (1,2,3), we have
∂f1
∂x = 2(1) = 2,∂f1
∂y = 2(2) = 4,∂f1
∂z = 0
∂f2
∂x = 2,∂f2
∂y = 3,∂f2
∂z = 2
∂f3
∂x = 0,∂f3
∂y = 0,∂f3
∂z = 27
Step 5: Therefore, the Jacobian matrix of fat (1,2,3) is
Jf=
2 4 0
2 3 2
0 0 27
Question 25
Question
Let f:R3→R2and g:R2→R3be defined by
f(x, y, z) = (xy, xz)
g(u, v)=(u+v, uv, u −v)
Find the Jacobian matrix of the composition function (g◦f)(x, y, z) and
evaluate it at the point (1,2,3).
21
Solution
Step 1: Calculate the composition function (g◦f)(x, y, z).
(g◦f)(x, y, z) = g(f(x, y, z)) = g(xy, xz)=(xy +xz, xy ·xz, xy −xz)
Step 2: Find the Jacobian matrix of (g◦f)(x, y, z). To find the Jacobian
matrix, we will compute the partial derivatives of each component of (g◦f)
with respect to x,y, and z.
Let u=xy +xz,v=xyz, and w=xy −xz. Then, we have
∂u
∂x =y+z
∂u
∂y =x
∂u
∂z =x
∂v
∂x =yz
∂v
∂y =xz
∂v
∂z =xy
∂w
∂x =y−z
∂w
∂y =x
∂w
∂z =−x
Therefore, the Jacobian matrix of (g◦f)(x, y, z) is
J=
y+z x x
yz xz xy
y−z x −x
Step 3: Evaluate the Jacobian matrix at the point (1, 2, 3). Substitute
x= 1, y= 2, and z= 3 into the Jacobian matrix Jto get
J(1,2,3) =
511
662
−1 1 −1
22
Question 26
Question
Find the Jacobian matrix Jof the transformation given by the equations
x=usin(v), y =ucos(v)
Solution
Step 1: Compute the partial derivatives of xand ywith respect to uand v.
∂x
∂u = sin(v),∂x
∂v =ucos(v),
∂y
∂u = cos(v),∂y
∂v =−usin(v).
Step 2: Assemble the Jacobian matrix using the partial derivatives.
J=∂x
∂u
∂x
∂v
∂y
∂u
∂y
∂v =sin(v)ucos(v)
cos(v)−usin(v)
Therefore, the Jacobian matrix Jof the transformation is:
J=sin(v)ucos(v)
cos(v)−usin(v)
Question 27
Question
Let f:R3→R2be a vector-valued function defined by f(x, y, z) = (x2+
yz, z2−xy). Find the Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to compute the partial derivatives of
each component function with respect to each variable.
Step 1: Find the partial derivatives of the first component f1(x, y, z) =
x2+yz.
∂f1
∂x = 2x,
∂f1
∂y =z,
∂f1
∂z =y.
23
Step 2: Find the partial derivatives of the second component f2(x, y, z) =
z2−xy.
∂f2
∂x =−y,
∂f2
∂y =−x,
∂f2
∂z = 2z.
Step 3: Assemble the partial derivatives into the Jacobian matrix Jf:
Jf="∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #=2x z y
−y−x2z.
Therefore, the Jacobian matrix of fis
Jf=2x z y
−y−x2z.
Question 28
Question
Let f(x, y) = (x2+y2, xy). Find the Jacobian matrix Jf(x, y) of the function
f.
Solution
To find the Jacobian matrix Jf(x, y) of the function f, we need to compute the
partial derivatives of fwith respect to each variable.
Step 1: Compute ∂f1
∂x and ∂f1
∂y :
∂f1
∂x =∂
∂x (x2+y2)=2x
∂f1
∂y =∂
∂y (x2+y2)=2y
Step 2: Compute ∂f2
∂x and ∂f2
∂y :
∂f2
∂x =∂
∂x (xy) = y
∂f2
∂y =∂
∂y (xy) = x
24
Step 3: Assemble the partial derivatives into the Jacobian matrix:
Jf(x, y) = "∂f1
∂x
∂f1
∂y
∂f2
∂x
∂f2
∂y #=2x2y
y x
Therefore, the Jacobian matrix of the function f(x, y) = (x2+y2, xy) is
2x2y
y x .
Question 29
Question
Let f(x, y, z)=(x2y+z, y2z, xyz) be a vector-valued function. Compute the
Jacobian matrix of f.
Solution
To compute the Jacobian matrix of f, we need to find the partial derivatives of
each component function with respect to each input variable.
Step 1: Find the partial derivatives of fwith respect to x.
The first component of fis f1(x, y, z) = x2y+z. Taking the partial derivative
of f1with respect to x, we get:
∂f1
∂x =∂
∂x (x2y+z)=2xy
Step 2: Find the partial derivatives of fwith respect to y.
The second component of fis f2(x, y, z) = y2z. Taking the partial derivative
of f2with respect to y, we have:
∂f2
∂y =∂
∂y (y2z)=2yz
Step 3: Find the partial derivatives of fwith respect to z.
The third component of fis f3(x, y, z) = xyz. Taking the partial derivative
of f3with respect to z, we obtain:
∂f3
∂z =∂
∂z (xyz) = xy
Step 4: Assemble the partial derivatives into the Jacobian matrix.
The Jacobian matrix of fwill be:
Jf(x, y, z) =
2xy x21
0 2yz 0
y x 0
Therefore, the Jacobian matrix of fis
2xy x21
0 2yz 0
y x 0
.
25
Question 30
Question
Let f:R3→R2be a function defined by f(x, y, z) = (x2y, z). Find the
Jacobian matrix of fat the point (1,1,1).
Solution
Step 1: Compute the partial derivatives of f(x, y, z).
∂f
∂x =∂
∂x (x2y)
∂
∂x (z)=2xy
0
∂f
∂y = ∂
∂y (x2y)
∂
∂y (z)!=x2
0
∂f
∂z =∂
∂z (x2y)
∂
∂z (z)=0
1
Step 2: Evaluate the partial derivatives at the point (1,1,1).
∂f
∂x (1,1,1) = 2·1·1
0=2
0
∂f
∂y (1,1,1) = 12
0=1
0
∂f
∂z (1,1,1) = 0
1
Step 3: Assemble the Jacobian matrix. The Jacobian matrix Jf(1,1,1) is
given by
Jf(1,1,1) = ∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z !=210
001
26
Question 2
Question
Let f(x, y, z) = (x2+y2, xy −z, z2) be a vector-valued function. Find the
Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to calculate the partial derivatives of
each component function.
Step 1: Calculate ∂f1
∂x ,∂f1
∂y , and ∂f1
∂z .
∂f1
∂x = 2x, ∂f1
∂y = 2y, ∂f1
∂z = 0
Step 2: Calculate ∂f2
∂x ,∂f2
∂y , and ∂f2
∂z .
∂f2
∂x =y, ∂f2
∂y =x, ∂f2
∂z =−1
Step 3: Calculate ∂f3
∂x ,∂f3
∂y , and ∂f3
∂z .
∂f3
∂x = 0,∂f3
∂y = 0,∂f3
∂z = 2z
Step 4: Assemble the Jacobian matrix Jf.
Jf=
2x2y0
y x −1
0 0 2z
Question 3
Question
Let f:R3→R2be a function defined by f(x, y, z) = (x2+yz, xyz). Determine
the Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to compute the partial derivatives of
each component function of fwith respect to the input variables.
Step 1: Find the partial derivatives of the first component function
2
Let g1(x, y, z) = x2+yz. We find the partial derivatives of g1as follows:
∂g1
∂x = 2x,
∂g1
∂y =z,
∂g1
∂z =y.
Step 2: Find the partial derivatives of the second component func-
tion
Let g2(x, y, z) = xyz. We find the partial derivatives of g2as follows:
∂g2
∂x =yz,
∂g2
∂y =xz,
∂g2
∂z =xy.
Step 3: Assemble the Jacobian matrix
The Jacobian matrix of fis the matrix of partial derivatives of the compo-
nent functions of f:
Jf= ∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z !=2x z y
yz xz xy.
Therefore, the Jacobian matrix of fis 2x z y
yz xz xy.
Question 4
Question
Let f:R3→R2be a function defined by f(x, y, z) = x2+yz, x
z. Determine
the Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to calculate the partial derivatives of
each component function.
Step 1: Find ∂f1
∂x ,∂f1
∂y , and ∂f1
∂z for the first component function f1(x, y, z) =
3
x2+yz.
∂f1
∂x = 2x,
∂f1
∂y =z,
∂f1
∂z =y.
Step 2: Find ∂f2
∂x ,∂f2
∂y , and ∂f2
∂z for the second component function f2(x, y, z) =
x
z.
∂f2
∂x =1
z,
∂f2
∂y = 0,
∂f2
∂z =−x
z2.
Step 3: Assemble the Jacobian matrix of fusing the partial derivatives
found in Step 1 and Step 2.
Jf="2x z y
1
z0−x
z2#.
Question 5
Question
Let f:R3→R2be a function defined by f(x, y, z) = (2x+y2, yz). Compute
the Jacobian matrix of f.
Solution
To compute the Jacobian matrix of f, we need to find the partial derivatives of
each component function with respect to each variable.
Step 1: Compute ∂f1∂x,∂f1
∂y , and ∂f1
∂z .
∂f1
∂x = 2,∂f1
∂y = 2y, ∂f1
∂z = 0
Step 2: Compute ∂f2∂x,∂f2
∂y , and ∂f2
∂z .
∂f2
∂x = 0,∂f2
∂y =z, ∂f2
∂z =y
4
Step 3: Assemble the Jacobian matrix. The Jacobian matrix of fis
given by
Jf="∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #
Substitute the partial derivatives we found in Steps 1 and 2 to get
Jf=2 2y0
0z y
Therefore, the Jacobian matrix of fis
Jf=2 2y0
0z y
Question 6
Question
Let f:R3→R2be a function given by:
f(x, y, z) = ex2+y2, z2
Compute the Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to compute the partial derivatives
of each component function with respect to the variables x,y, and z, and then
arrange them in a matrix.
Step 1: Find ∂f1
∂x ,∂f1
∂y , and ∂f1
∂z . Starting with the first component function:
f1(x, y, z) = ex2+y2
We have: ∂f1
∂x = 2xex2+y2
∂f1
∂y = 2yex2+y2
∂f1
∂z = 0
Step 2: Find ∂f2
∂x ,∂f2
∂y , and ∂f2
∂z . Moving on to the second component
function:
f2(x, y, z) = z2
We have: ∂f2
∂x = 0
5
∂f2
∂y = 0
∂f2
∂z = 2z
Step 3: Assemble the Jacobian matrix. The Jacobian matrix of fis given
by:
Jacobian(f) = "∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #
Substituting the partial derivatives we found:
Jacobian(f) = 2xex2+y22yex2+y20
0 0 2z
Question 7
Question
Let f:R3→R2be defined by f(x, y, z)=(x2+yz, y2−xz). Find the Jacobian
matrix of fat the point (1,2,−1).
Solution
Step 1: Calculate the partial derivatives of fwith respect to x,y, and z.
∂f1
∂x =∂
∂x (x2+yz)=2x
∂f1
∂y =∂
∂y (x2+yz) = z
∂f1
∂z =∂
∂z (x2+yz) = y
∂f2
∂x =∂
∂x (y2−xz) = −z
∂f2
∂y =∂
∂y (y2−xz)=2y
∂f2
∂z =∂
∂z (y2−xz) = −x
6
Step 2: Evaluate the partial derivatives at the point (1,2,−1).
∂f1
∂x (1,2,−1) = 2(1) = 2
∂f1
∂y (1,2,−1) = (−1) = −1
∂f1
∂z (1,2,−1) = 2
∂f2
∂x (1,2,−1) = 1
∂f2
∂y (1,2,−1) = 4
∂f2
∂z (1,2,−1) = −1
Step 3: Assemble the Jacobian matrix.
Jf(1,2,−1) = "∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #=2−1 2
1 4 −1
Question 8
Question
Let f(x, y, z)=(x2+y, 2y−z, 3z+x) be a vector-valued function in R3. Find
the Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to compute the partial derivatives of
the components of fwith respect to x,y, and z.
Step 1: Partial derivative of f1with respect to x:
∂f1
∂x =∂(x2+y)
∂x = 2x
Step 2: Partial derivative of f1with respect to y:
∂f1
∂y =∂(x2+y)
∂y = 1
Step 3: Partial derivative of f1with respect to z:
∂f1
∂z =∂(x2+y)
∂z = 0
Step 4: Partial derivative of f2with respect to x:
∂f2
∂x =∂(2y−z)
∂x = 0
7
Step 5: Partial derivative of f2with respect to y:
∂f2
∂y =∂(2y−z)
∂y = 2
Step 6: Partial derivative of f2with respect to z:
∂f2
∂z =∂(2y−z)
∂z =−1
Step 7: Partial derivative of f3with respect to x:
∂f3
∂x =∂(3z+x)
∂x = 1
Step 8: Partial derivative of f3with respect to y:
∂f3
∂y =∂(3z+x)
∂y = 0
Step 9: Partial derivative of f3with respect to z:
∂f3
∂z =∂(3z+x)
∂z = 3
Therefore, the Jacobian matrix of fis:
J(f) =
2 1 0
0 2 −1
1 0 3
Question 9
Question
Find the Jacobian matrix of the following function:
f(x, y, z)=(xy, x2z, y2z)
Solution
To find the Jacobian matrix of the function f(x, y, z) = (xy, x2z, y2z), we need
to compute the partial derivatives of each component function with respect to
each variable.
Step 1: Calculate the partial derivatives of fwith respect to x:
∂f1
∂x =∂
∂x (xy) = y
∂f2
∂x =∂
∂x (x2z)=2xz
∂f3
∂x =∂
∂x (y2z)=0
8
Step 2: Calculate the partial derivatives of fwith respect to y:
∂f1
∂y =∂
∂y (xy) = x
∂f2
∂y =∂
∂y (x2z)=0
∂f3
∂y =∂
∂y (y2z)=2yz
Step 3: Calculate the partial derivatives of fwith respect to z:
∂f1
∂z =∂
∂z (xy) = 0
∂f2
∂z =∂
∂z (x2z) = x2
∂f3
∂z =∂
∂z (y2z) = y2
Therefore, the Jacobian matrix of f(x, y, z)=(xy, x2z, y2z) is:
J=
y x 0
2xz 0 2yz
0x2y2
Question 10
Question
Let f:R3→R3be a mapping defined by f(x, y, z)=(xy, xz, yz). Determine
the Jacobian matrix of f.
Solution
To determine the Jacobian matrix of f, we need to compute the partial deriva-
tives of each component of fwith respect to each input variable.
Step 1: Calculate the partial derivatives of f:
∂f1
∂x =∂(xy)
∂x =y, ∂f1
∂y =∂(xy)
∂y =x, ∂f1
∂z = 0
∂f2
∂x =∂(xz)
∂x =z, ∂f2
∂y = 0,∂f2
∂z =∂(xz)
∂z =x
∂f3
∂x = 0,∂f3
∂y =∂(yz)
∂y =z, ∂f3
∂z =∂(yz)
∂z =y
Step 2: Assemble the Jacobian matrix: The Jacobian matrix of fis given
by
J(f) =
∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z
∂f3
∂x
∂f3
∂y
∂f3
∂z
=
y x 0
z0x
0z y
9
Therefore, the Jacobian matrix of fis
y x 0
z0x
0z y
.
Question 11
Question
Let f:R3→R2be a function defined by f(x, y, z)=(x2y, yz). Find the
Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to calculate the partial derivatives of
each component function with respect to each variable.
Step 1: Calculate ∂f1
∂x ,∂f1
∂y , and ∂f1
∂z for the first component function
f1(x, y, z) = x2y.
∂f1
∂x = 2xy
∂f1
∂y =x2
∂f1
∂z = 0
Step 2: Calculate ∂f2
∂x ,∂f2
∂y , and ∂f2
∂z for the second component function
f2(x, y, z) = yz.
∂f2
∂x = 0
∂f2
∂y =z
∂f2
∂z =y
Step 3: Assemble the partial derivatives into a matrix to form the Jacobian
matrix of f.
Jf=2xy x20
0z y
Question 12
Question
Let f:R3→R2be a function defined by f(x, y, z)=(xy2z, x2z). Find the
Jacobian matrix of fat the point (2,−1,3).
10
Solution
Step 1: Compute the partial derivatives of fwith respect to x, y, and z.
∂f
∂x =y2z, 2xz
∂f
∂y = (2xyz, 0)
∂f
∂z =xy2, x2
Step 2: Evaluate the partial derivatives at the point (2,−1,3).
∂f
∂x (2,−1,3) = (−1)2(3),2(2)(3)= (3,12)
∂f
∂y (2,−1,3) = (2(2)(−1)(3),0) = (−12,0)
∂f
∂z (2,−1,3) = 2(−1)2,22= (−2,4)
Step 3: Form the Jacobian matrix by arranging the partial derivatives as
rows.
Jf(2,−1,3) =
3 12
−12 0
−2 4
Question 13
Question
Let f:R3→R3be a vector field given by f(x, y, z)=(xy, xz, yz2). Find the
Jacobian matrix of f.
Solution
The Jacobian matrix of a vector field f:Rn→Rmis an m×nmatrix where
each entry is the partial derivative of the i-th component of fwith respect to
the j-th variable.
Step 1: Find the Jacobian matrix Jfof f. The components of fare
f1(x, y, z) = xy,f2(x, y, z) = xz, and f3(x, y, z) = yz2. We need to compute
the partial derivatives of each component with respect to each variable:
∂f1
∂x =y, ∂f1
∂y =x, ∂f1
∂z = 0,
∂f2
∂x =z, ∂f2
∂y = 0,∂f2
∂z =x,
∂f3
∂x = 0,∂f3
∂y =z2,∂f3
∂z = 2yz.
11
Therefore, the Jacobian matrix Jfis:
Jf=
∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z
∂f3
∂x
∂f3
∂y
∂f3
∂z
=
y x 0
z0x
0z22yz
.
Therefore, the Jacobian matrix of fis
y x 0
z0x
0z22yz
.
Question 14
Question
Let f(x, y, z)=(x2+y2, xy, xz) be a vector-valued function. Determine the
Jacobian matrix Jfof f.
Solution
To find the Jacobian matrix Jfof a vector-valued function f(x, y, z), we need
to compute the partial derivatives of each component function with respect to
each variable.
Step 1: Find the partial derivatives of fwith respect to x.
∂f
∂x =
∂
∂x (x2+y2)
∂
∂x (xy)
∂
∂x (xz)
=
2x
y
z
Step 2: Find the partial derivatives of fwith respect to y.
∂f
∂y =
∂
∂y (x2+y2)
∂
∂y (xy)
∂
∂y (xz)
=
2y
x
0
Step 3: Find the partial derivatives of fwith respect to z.
∂f
∂z =
∂
∂z (x2+y2)
∂
∂z (xy)
∂
∂z (xz)
=
0
0
x
Step 4: Assemble the partial derivatives into the Jacobian matrix Jf.
Jf=
2x2y0
y x 0
z0x
12
Question 15
Question
Let f(x, y, z) = x2+y2+z2and g(x, y, z) = exy . Find the Jacobian matrix of
the function g◦f, where g◦frepresents the composition of functions.
Solution
Step 1: Find the composition of functions g◦f(x, y, z).
g◦f(x, y, z) = g(f(x, y, z)) = e(x2+y2+z2)y
Step 2: Calculate the partial derivatives of g◦fwith respect to x, y, and z.
∂
∂x e(x2+y2+z2)y= 2xy2e(x2+y2+z2)y
∂
∂y e(x2+y2+z2)y= (x2+ 2y2+z2)e(x2+y2+z2)y
∂
∂z e(x2+y2+z2)y= 2yze(x2+y2+z2)y
Step 3: Assemble the Jacobian matrix of g◦f.
J(g◦f) = h2xy2e(x2+y2+z2)y(x2+ 2y2+z2)e(x2+y2+z2)y2yze(x2+y2+z2)yi
Therefore, the Jacobian matrix of g◦fis:
J(g◦f) = h2xy2e(x2+y2+z2)y(x2+ 2y2+z2)e(x2+y2+z2)y2yze(x2+y2+z2)yi
Question 16
Question
Let f:R3→R2be a differentiable function defined by f(x, y, z)=(x2y, yz).
Find the Jacobian matrix of f.
Solution
Step 1: The Jacobian matrix of fis given by the matrix of all first-order partial
derivatives of f, denoted by Jf.
Step 2: To find the Jacobian matrix of f, we need to compute the partial
derivatives of the components of fwith respect to x,y, and z.
Step 3: The components of fare f1(x, y, z) = x2yand f2(x, y, z) = yz.
13
Step 4: Let’s compute the partial derivatives of f1with respect to x,y, and
z:∂f1
∂x = 2xy,
∂f1
∂y =x2,
∂f1
∂z = 0.
Step 5: Now, let’s compute the partial derivatives of f2with respect to x,
y, and z:
∂f2
∂x = 0,
∂f2
∂y =z,
∂f2
∂z =y.
Step 6: The Jacobian matrix Jfis formed by these partial derivatives:
Jf=2xy x20
0z y.
Therefore, the Jacobian matrix of fis 2xy x20
0z y.
Question 17
Question
Let f:R3→R3be a vector field given by f(x, y, z)=(x2y, z, y2). Compute
the Jacobian matrix of f.
Solution
The Jacobian matrix of a vector field f:Rn→Rmis an m×nmatrix whose
entries are the partial derivatives of the components of f.
14
Step 1: Find the partial derivatives of the components of f.
∂
∂x (x2y)=2xy
∂
∂y (x2y) = x2
∂
∂z (x2y)=0
∂
∂x (z)=0
∂
∂y (z)=0
∂
∂z (z)=1
∂
∂x (y2)=0
∂
∂y (y2)=2y
∂
∂z (y2)=0
Step 2: Assemble the partial derivatives into the Jacobian matrix.
Jf=
2xy x20
0 0 1
0 2y0
Question 18
Question
Find the Jacobian matrix of the following function:
f(x, y, z) =
x2y
yz
x+z
Solution
To find the Jacobian matrix of a vector-valued function, we need to compute
the partial derivatives of each component function with respect to each variable.
Step 1: Compute the partial derivatives of fwith respect to x,y, and z.
Let’s start by finding the partial derivative of the first component function
x2ywith respect to x:
∂(x2y)
∂x = 2xy
15
Next, find the partial derivative of x2ywith respect to y:
∂(x2y)
∂y =x2
Moving on to the second component function yz, find the partial derivative
with respect to y:
∂(yz)
∂y =z
Finally, find the partial derivative of yz with respect to z:
∂(yz)
∂z =y
For the last component function x+z, find the partial derivative with respect
to x:∂(x+z)
∂x = 1
Next, find the partial derivative of x+zwith respect to z:
∂(x+z)
∂z = 1
Step 2: Assemble the partial derivatives into the Jacobian matrix.
Putting all the partial derivatives together, the Jacobian matrix of f(x, y, z)
is:
Jf(x, y, z) =
2xy x20
0z y
1 0 1
Question 19
Question
Consider the following system of equations:
(x2+y2= 25
xy = 9
Find the Jacobian matrix of the system at the point where x= 3 and y= 3.
Solution
To find the Jacobian matrix of the system at the given point, we first need
to find the partial derivatives of the functions F(x, y) = x2+y2−25 and
G(x, y) = xy −9.
Step 1: Find partial derivatives
16
For F(x, y) = x2+y2−25:
∂F
∂x = 2x
∂F
∂y = 2y
For G(x, y) = xy −9:
∂G
∂x =y
∂G
∂y =x
Step 2: Evaluate partial derivatives at the given point At the point
where x= 3 and y= 3:
∂F
∂x = 2(3) = 6
∂F
∂y = 2(3) = 6
∂G
∂x = 3
∂G
∂y = 3
Step 3: Assemble the Jacobian matrix The Jacobian matrix is given
by:
J(x, y) = "∂F
∂x
∂F
∂y
∂G
∂x
∂G
∂y #
Substitute the evaluated partial derivatives at the point x= 3 and y= 3
into the Jacobian matrix:
J(3,3) = 6 6
3 3
Therefore, the Jacobian matrix of the system at the point where x= 3 and
y= 3 is:
6 6
3 3
Question 20
Question
Let f:R2→R2be a function defined by f(x, y) = (x2+y2, xy). Find the
Jacobian matrix of fat the point (1,2).
17
Solution
Step 1: Compute the partial derivatives of fwith respect to xand y.
∂f1
∂x =∂
∂x (x2+y2)=2x,
∂f1
∂y =∂
∂y (x2+y2)=2y,
∂f2
∂x =∂
∂x (xy) = y,
∂f2
∂y =∂
∂y (xy) = x.
Step 2: Evaluate the partial derivatives at the point (1,2).
∂f1
∂x (1,2)
= 2(1) = 2,
∂f1
∂y (1,2)
= 2(2) = 4,
∂f2
∂x (1,2)
= 2,
∂f2
∂y (1,2)
= 1.
Step 3: Assemble the Jacobian matrix at (1,2). The Jacobian matrix of f
at (1,2) is given by
Jf(1,2) = 2 4
2 1.
Question 21
Question
Let f:R3→R3be defined by f(x, y, z) = (x2+yz, yz −x2, x +y+z). Find
the Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to compute the partial derivatives of
each component function.
Step 1: Compute the partial derivatives of fwith respect to x.
∂f1
∂x = 2x, ∂f2
∂x =−2x, ∂f3
∂x = 1
18
Step 2: Compute the partial derivatives of fwith respect to y.
∂f1
∂y =z, ∂f2
∂y =z, ∂f3
∂y = 1
Step 3: Compute the partial derivatives of fwith respect to z.
∂f1
∂z =y, ∂f2
∂z =y, ∂f3
∂z = 1
Step 4: Assemble the partial derivatives into the Jacobian matrix:
Jf=
∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z
∂f3
∂x
∂f3
∂y
∂f3
∂z
=
2x z y
−2x z y
1 1 1
Question 22
Question
Let f:R3→R2be a function defined by f(x, y, z) = (2x+y, 3y−z). Determine
the Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to compute the partial derivatives of
each component function with respect to each variable.
Step 1: Compute the partial derivatives of f: Let f(x, y, z)=(u, v) where
u= 2x+yand v= 3y−z. The partial derivatives are:
∂u
∂x = 2,∂u
∂y = 1,∂u
∂z = 0
∂v
∂x = 0,∂v
∂y = 3,∂v
∂z =−1
Step 2: Assemble the Jacobian matrix: The Jacobian matrix of fis a 2 ×3
matrix where each row corresponds to a component function and each column
corresponds to a variable. In this case, the Jacobian matrix is:
"∂u
∂x
∂u
∂y
∂u
∂z
∂v
∂x
∂v
∂y
∂v
∂z #=2 1 0
0 3 −1
Therefore, the Jacobian matrix of fis 2 1 0
0 3 −1.
19
Question 23
Question
Let f:R3→R2be defined by f(x, y, z) = (xy, yz). Calculate the Jacobian
matrix of f.
Solution
To find the Jacobian matrix of f, we need to calculate the partial derivatives of
each component of fwith respect to each input variable.
Step 1: Find ∂f1
∂x ,∂f1
∂y , and ∂f1
∂z .
Given f(x, y, z) = (xy, yz), we have:
∂f1
∂x =y
∂f1
∂y =x
∂f1
∂z = 0
Step 2: Find ∂f2
∂x ,∂f2
∂y , and ∂f2
∂z .
Given f(x, y, z) = (xy, yz), we have:
∂f2
∂x = 0
∂f2
∂y =z
∂f2
∂z =y
Step 3: Assemble the Jacobian matrix.
The Jacobian matrix of fis given by:
Jf="∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #
Substitute the partial derivatives we found into the Jacobian matrix:
Jf=y x 0
0z y
Question 24
Question
Let f(x, y, z)=(x2+y2, xyz, z3). Find the Jacobian matrix of fat the point
(1,2,3).
20
Solution
Step 1: First, we need to calculate the partial derivatives of fwith respect to
each variable x,y, and z.
Step 2: The Jacobian matrix of fis given by
Jf=
∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z
∂f3
∂x
∂f3
∂y
∂f3
∂z
Step 3: Calculating the partial derivatives, we get
∂f1
∂x = 2x, ∂f1
∂y = 2y, ∂f1
∂z = 0
∂f2
∂x =yz, ∂f2
∂y =xz, ∂f2
∂z =xy
∂f3
∂x = 0,∂f3
∂y = 0,∂f3
∂z = 3z2
Step 4: Evaluating the partial derivatives at the point (1,2,3), we have
∂f1
∂x = 2(1) = 2,∂f1
∂y = 2(2) = 4,∂f1
∂z = 0
∂f2
∂x = 2,∂f2
∂y = 3,∂f2
∂z = 2
∂f3
∂x = 0,∂f3
∂y = 0,∂f3
∂z = 27
Step 5: Therefore, the Jacobian matrix of fat (1,2,3) is
Jf=
2 4 0
2 3 2
0 0 27
Question 25
Question
Let f:R3→R2and g:R2→R3be defined by
f(x, y, z) = (xy, xz)
g(u, v)=(u+v, uv, u −v)
Find the Jacobian matrix of the composition function (g◦f)(x, y, z) and
evaluate it at the point (1,2,3).
21
Solution
Step 1: Calculate the composition function (g◦f)(x, y, z).
(g◦f)(x, y, z) = g(f(x, y, z)) = g(xy, xz)=(xy +xz, xy ·xz, xy −xz)
Step 2: Find the Jacobian matrix of (g◦f)(x, y, z). To find the Jacobian
matrix, we will compute the partial derivatives of each component of (g◦f)
with respect to x,y, and z.
Let u=xy +xz,v=xyz, and w=xy −xz. Then, we have
∂u
∂x =y+z
∂u
∂y =x
∂u
∂z =x
∂v
∂x =yz
∂v
∂y =xz
∂v
∂z =xy
∂w
∂x =y−z
∂w
∂y =x
∂w
∂z =−x
Therefore, the Jacobian matrix of (g◦f)(x, y, z) is
J=
y+z x x
yz xz xy
y−z x −x
Step 3: Evaluate the Jacobian matrix at the point (1, 2, 3). Substitute
x= 1, y= 2, and z= 3 into the Jacobian matrix Jto get
J(1,2,3) =
511
662
−1 1 −1
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Question 26
Question
Find the Jacobian matrix Jof the transformation given by the equations
x=usin(v), y =ucos(v)
Solution
Step 1: Compute the partial derivatives of xand ywith respect to uand v.
∂x
∂u = sin(v),∂x
∂v =ucos(v),
∂y
∂u = cos(v),∂y
∂v =−usin(v).
Step 2: Assemble the Jacobian matrix using the partial derivatives.
J=∂x
∂u
∂x
∂v
∂y
∂u
∂y
∂v =sin(v)ucos(v)
cos(v)−usin(v)
Therefore, the Jacobian matrix Jof the transformation is:
J=sin(v)ucos(v)
cos(v)−usin(v)
Question 27
Question
Let f:R3→R2be a vector-valued function defined by f(x, y, z) = (x2+
yz, z2−xy). Find the Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to compute the partial derivatives of
each component function with respect to each variable.
Step 1: Find the partial derivatives of the first component f1(x, y, z) =
x2+yz.
∂f1
∂x = 2x,
∂f1
∂y =z,
∂f1
∂z =y.
23
Step 2: Find the partial derivatives of the second component f2(x, y, z) =
z2−xy.
∂f2
∂x =−y,
∂f2
∂y =−x,
∂f2
∂z = 2z.
Step 3: Assemble the partial derivatives into the Jacobian matrix Jf:
Jf="∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #=2x z y
−y−x2z.
Therefore, the Jacobian matrix of fis
Jf=2x z y
−y−x2z.
Question 28
Question
Let f(x, y) = (x2+y2, xy). Find the Jacobian matrix Jf(x, y) of the function
f.
Solution
To find the Jacobian matrix Jf(x, y) of the function f, we need to compute the
partial derivatives of fwith respect to each variable.
Step 1: Compute ∂f1
∂x and ∂f1
∂y :
∂f1
∂x =∂
∂x (x2+y2)=2x
∂f1
∂y =∂
∂y (x2+y2)=2y
Step 2: Compute ∂f2
∂x and ∂f2
∂y :
∂f2
∂x =∂
∂x (xy) = y
∂f2
∂y =∂
∂y (xy) = x
24
Step 3: Assemble the partial derivatives into the Jacobian matrix:
Jf(x, y) = "∂f1
∂x
∂f1
∂y
∂f2
∂x
∂f2
∂y #=2x2y
y x
Therefore, the Jacobian matrix of the function f(x, y) = (x2+y2, xy) is
2x2y
y x .
Question 29
Question
Let f(x, y, z)=(x2y+z, y2z, xyz) be a vector-valued function. Compute the
Jacobian matrix of f.
Solution
To compute the Jacobian matrix of f, we need to find the partial derivatives of
each component function with respect to each input variable.
Step 1: Find the partial derivatives of fwith respect to x.
The first component of fis f1(x, y, z) = x2y+z. Taking the partial derivative
of f1with respect to x, we get:
∂f1
∂x =∂
∂x (x2y+z)=2xy
Step 2: Find the partial derivatives of fwith respect to y.
The second component of fis f2(x, y, z) = y2z. Taking the partial derivative
of f2with respect to y, we have:
∂f2
∂y =∂
∂y (y2z)=2yz
Step 3: Find the partial derivatives of fwith respect to z.
The third component of fis f3(x, y, z) = xyz. Taking the partial derivative
of f3with respect to z, we obtain:
∂f3
∂z =∂
∂z (xyz) = xy
Step 4: Assemble the partial derivatives into the Jacobian matrix.
The Jacobian matrix of fwill be:
Jf(x, y, z) =
2xy x21
0 2yz 0
y x 0
Therefore, the Jacobian matrix of fis
2xy x21
0 2yz 0
y x 0
.
25
Question 30
Question
Let f:R3→R2be a function defined by f(x, y, z) = (x2y, z). Find the
Jacobian matrix of fat the point (1,1,1).
Solution
Step 1: Compute the partial derivatives of f(x, y, z).
∂f
∂x =∂
∂x (x2y)
∂
∂x (z)=2xy
0
∂f
∂y = ∂
∂y (x2y)
∂
∂y (z)!=x2
0
∂f
∂z =∂
∂z (x2y)
∂
∂z (z)=0
1
Step 2: Evaluate the partial derivatives at the point (1,1,1).
∂f
∂x (1,1,1) = 2·1·1
0=2
0
∂f
∂y (1,1,1) = 12
0=1
0
∂f
∂z (1,1,1) = 0
1
Step 3: Assemble the Jacobian matrix. The Jacobian matrix Jf(1,1,1) is
given by
Jf(1,1,1) = ∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z !=210
001
26
