MATH 117 - ELEMENTS OF
MATHEMATICS - Jacobian matrix
Question Bank - Set 8
Liberty University
Question 1
Question
Let f:R3→R2be a differentiable function given by f(x, y, z)=(x2+y, xyz).
Calculate 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 the partial derivatives of f: Let f(x, y, z)=(u, v), where
u=x2+yand v=xyz. The partial derivatives of uand vwith respect to x,
y, and zare: ∂u
∂x = 2x, ∂u
∂y = 1,∂u
∂z = 0
∂v
∂x =yz, ∂v
∂y =xz, ∂v
∂z =xy
Step 2: Assemble the Jacobian matrix: The Jacobian matrix of fis given
by
Jf= ∂u
∂x
∂u
∂y
∂u
∂z
∂v
∂x
∂v
∂y
∂v
∂z !
Substitute the partial derivatives we found earlier into the matrix:
Jf=2x1 0
yz xz xy
So, the Jacobian matrix of fis 2x1 0
yz xz xy.
Question 2
Question
Find the Jacobian matrix of the following system of equations:
(x=u2+v2
y=uv
Solution
Step 1: Compute the partial derivatives of xand ywith respect to uand v.
For x=u2+v2:∂x
∂u = 2uand ∂x
∂v = 2v
For y=uv:∂y
∂u =vand ∂y
∂v =u
Step 2: Assemble the partial derivatives into the Jacobian matrix. The
Jacobian matrix is given by:
J=∂x
∂u
∂x
∂v
∂y
∂u
∂y
∂v
Substitute the computed partial derivatives:
J=2u2v
v u
Therefore, the Jacobian matrix of the given system of equations is:
J=2u2v
v u
Question 3
Question
Let f:R3→R2be defined by f(x, y, z)=(x2yz, sin(yz)). Find the Jacobian
matrix of fat the point (1,0, π).
2
Solution
Step 1: Compute the partial derivatives of fwith respect to x,y, and z.
∂f1
∂x =∂
∂x(x2yz) = 2xyz,
∂f1
∂y =∂
∂y (x2yz) = x2z,
∂f1
∂z =∂
∂z (x2yz) = x2y,
∂f2
∂x =∂
∂x(sin(yz)) = 0,
∂f2
∂y =∂
∂y (sin(yz)) = zcos(yz),
∂f2
∂z =∂
∂z (sin(yz)) = ycos(yz).
Step 2: Evaluate the partial derivatives at the point (1,0, π).
∂f
∂x (1,0,π)
= 2(1)(0)(π) = 0,
∂f
∂y (1,0,π)
= (12)(π) = π,
∂f
∂z (1,0,π)
= (12)(0) = 0,
∂f
∂x (1,0,π)
= 0,
∂f
∂y (1,0,π)
= 0,
∂f
∂z (1,0,π)
= 0.
Step 3: Assemble the Jacobian matrix of fat the point (1,0, π).
Jacobian Matrix = 0π0
000.
Question 4
Question
Let f:R3→R3be a function defined by f(x, y, z) = (x2y+yz, xz, xyz).
Calculate the Jacobian matrix of f.
3
Solution
To find the Jacobian matrix of f, we need to calculate the partial derivatives of
each component function.
Step 1: Calculating the partial derivatives of fLet’s find the partial
derivatives of fwith respect to each variable.
For the first component function f1(x, y, z) = x2y+yz:
∂f1
∂x = 2xy, ∂f1
∂y =x2+z, ∂f1
∂z =y
For the second component function f2(x, y, z) = xz:
∂f2
∂x =z, ∂f2
∂y = 0,∂f2
∂z =x
For the third component function f3(x, y, z) = xyz:
∂f3
∂x =yz, ∂f3
∂y =xz, ∂f3
∂z =xy
Step 2: Constructing the Jacobian matrix The Jacobian matrix of f
is given by:
Jf=
∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z
∂f3
∂x
∂f3
∂y
∂f3
∂z
Substitute the partial derivatives we found into the Jacobian matrix:
Jf=
2xy x2+z y
z0x
yz xz xy
Therefore, the Jacobian matrix of fis:
Jf=
2xy x2+z y
z0x
yz xz xy
Question 5
Question
Let f:R2→R2be a mapping defined by f(x, y) = (x2+y2, xy). Compute the
Jacobian matrix of f.
4
Solution
To find the Jacobian matrix of f, we need to compute the partial derivatives of
each component function.
Step 1: Compute the partial derivative of the first component function with
respect to x.∂f1
∂x =∂
∂x(x2+y2)=2x
Step 2: Compute the partial derivative of the first component function with
respect to y.∂f1
∂y =∂
∂y (x2+y2)=2y
Step 3: Compute the partial derivative of the second component function
with respect to x.∂f2
∂x =∂
∂x(xy) = y
Step 4: Compute the partial derivative of the second component function
with respect to y.∂f2
∂y =∂
∂y (xy) = x
Step 5: Assemble the partial derivatives into the Jacobian matrix. The
Jacobian matrix of fis given by:
"∂f1
∂x
∂f1
∂y
∂f2
∂x
∂f2
∂y #=2x2y
y x
Question 6
Question
Let f(x, y)=(xy2, x2y) be a vector-valued function in R2. Determine the
Jacobian matrix of f(x, y).
Solution
To find the Jacobian matrix of f(x, y), we will compute the partial derivatives
of each component function.
Step 1: Find ∂f1
∂x and ∂f1
∂y .
∂f1
∂x =y2and ∂f1
∂y = 2xy
Step 2: Find ∂f2
∂x and ∂f2
∂y .
∂f2
∂x = 2xy and ∂f2
∂y =x2
5
Step 3: Assemble the Jacobian matrix. The Jacobian matrix of f(x, y) is
given by
Jf(x, y) = "∂f1
∂x
∂f1
∂y
∂f2
∂x
∂f2
∂y #
Substitute the partial derivatives we found into the matrix:
Jf(x, y) = y22xy
2xy x2
Therefore, the Jacobian matrix of f(x, y)=(xy2, x2y) is
Jf(x, y) = y22xy
2xy x2
Question 7
Question
Let f:R3→R2be a function defined by f(x, y, z) = (2x2y+z, xz2). Calculate
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:
Jf="∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #
Step 2: Let’s find the partial derivatives of fwith respect to x,y, and z.
∂f1
∂x = 4xy
∂f1
∂y = 2x2
∂f1
∂z = 1
Step 3: Similarly, ∂f2
∂x =z2
∂f2
∂y = 0
∂f2
∂z = 2xz
Step 4: Putting all the partial derivatives together, we get the Jacobian
matrix Jf:
Jf=4xy 2x21
z20 2xz
6
Question 8
Question
Let f:R3→R2be a function defined by f(x, y, z)=(x2+yz, xz). Find the
Jacobian matrix of f.
Solution
Step 1: The Jacobian matrix of a function f:Rn→Rmis an m×nmatrix
whose entries are the partial derivatives of the component functions of f. In this
case, the function f:R3→R2has component functions f1(x, y, z) = x2+yz
and f2(x, y, z) = xz.
Step 2: To find the Jacobian matrix of f, we first compute the partial
derivatives of f1and f2with respect to x,y, and z:
∂f1
∂x = 2x, ∂f1
∂y =z, ∂f1
∂z =y
∂f2
∂x =z, ∂f2
∂y = 0,∂f2
∂z =x
Step 3: Now, we construct the Jacobian matrix of fby arranging these
partial derivatives as rows:
Jf(x, y, z) = 2x z y
z0x
Therefore, the Jacobian matrix of fis given by
Jf(x, y, z) = 2x z y
z0x
Question 9
Question
Let f:R3→R2be a differentiable function defined by f(x, y, z)=(xy+yz, xz).
Find the Jacobian matrix of fat the point (1,2,3).
Solution
Step 1: Compute the partial derivatives of fwith respect to x,y, and z.
∂f1
∂x =∂
∂x(xy +yz) = y,
∂f1
∂y =∂
∂y (xy +yz) = x+z,
∂f1
∂z =∂
∂z (xy +yz) = y;
7
∂f2
∂x =∂
∂x(xz) = z,
∂f2
∂y =∂
∂y (xz)=0,
∂f2
∂z =∂
∂z (xz) = x.
Step 2: Evaluate the partial derivatives at the point (1,2,3).
∂f1
∂x (1,2,3) = 2,
∂f1
∂y (1,2,3) = 1 + 3 = 4,
∂f1
∂z (1,2,3) = 2;
∂f2
∂x (1,2,3) = 3,
∂f2
∂y (1,2,3) = 0,
∂f2
∂z (1,2,3) = 1.
Step 3: Assemble the Jacobian matrix of fat (1,2,3). The Jacobian matrix
of fat (1,2,3) is
Jf(1,2,3) = 242
301.
Question 10
Question
Let f(x, y, z) = x2y+yz3−sin(z). Find the Jacobian matrix of fat the point
(1,2, π).
Solution
To find the Jacobian matrix of fat the point (1,2, π), we need to compute the
partial derivatives of fwith respect to each variable x,y, and zat that point.
Step 1: Find ∂f ∂x
∂f
∂x =∂
∂x(x2y+yz3−sin(z)) = 2xy
Step 2: Find ∂f ∂y
∂f
∂y =∂
∂y (x2y+yz3−sin(z)) = x2+z3
8
Step 3: Find ∂f ∂z
∂f
∂z =∂
∂z (x2y+yz3−sin(z)) = 3yz2−cos(z)
Step 4: Evaluate the Jacobian matrix at (1,2, π) Now, substitute
x= 1, y= 2, and z=πinto the partial derivatives we found earlier:
∂f
∂x (1,2, π) = 2(1)(2) = 4
∂f
∂y (1,2, π)=12+ (π)3= 1 + π3
∂f
∂z (1,2, π) = 3(2)(π)2−cos(π)=6π2+ 1
Therefore, the Jacobian matrix of fat the point (1,2, π) is:
Jf(1,2, π) = 4 1 + π36π2+ 1
Question 11
Question
Let f(x, y, z)=(x2−yz, y3−xz, z2−xy). Find the Jacobian matrix of fwith
respect to x,y, and z.
Solution
Step 1: Compute the partial derivatives of fwith respect to x,y, and z.
∂f
∂x =
∂f1
∂x
∂f2
∂x
∂f3
∂x
=
2x−z−y
−z0y
−y−x2z
∂f
∂y =
∂f1
∂y
∂f2
∂y
∂f3
∂y
=
−z3y2−x
0 3y z
−x−z2y
∂f
∂z =
∂f1
∂z
∂f2
∂z
∂f3
∂z
=
−y0x
−x0z
2z−y−x
Step 2: Arrange the partial derivatives in a matrix to form the Jacobian
matrix of f.
Jf=
2x−z−y
−z0y
−y−x2z
,
−z3y2−x
0 3y z
−x−z2y
,
−y0x
−x0z
2z−y−x
9
Question 12
Question
Let f:R3→R2be a function defined by f(x, y, z)=(x2+yz, xz). 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.
Step 1: Compute ∂f1∂x,∂f1
∂y , and ∂f1
∂z
∂f1
∂x = 2x
∂f1
∂y =z
∂f1
∂z =y
Step 2: Compute ∂f2∂x,∂f2
∂y , and ∂f2
∂z
∂f2
∂x =z
∂f2
∂y = 0
∂f2
∂z =x
Step 3: Assemble the Jacobian matrix The Jacobian matrix of fis
given by
"∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #
Plugging in the computed partial derivatives, we have
2x z y
z0x
Question 13
Question
Find the Jacobian matrix of the transformation defined by x=rcos(θ) and
y=rsin(θ).
10
Solution
To find the Jacobian matrix of the transformation, we first need to express the
transformation in terms of rand θ.
Step 1: Express the transformation in terms of rand θ.
x=rcos(θ)
y=rsin(θ)
Step 2: Calculate the partial derivatives of the transformation with respect
to rand θ.∂x
∂r = cos(θ),∂x
∂θ =−rsin(θ)
∂y
∂r = sin(θ),∂y
∂θ =rcos(θ)
Step 3: Assemble the Jacobian matrix. The Jacobian matrix is given by:
J=∂x
∂r
∂x
∂θ
∂y
∂r
∂y
∂θ
Step 4: Substitute the partial derivatives into the Jacobian matrix.
J=cos(θ)−rsin(θ)
sin(θ)rcos(θ)
Therefore, the Jacobian matrix of the transformation is:
J=cos(θ)−rsin(θ)
sin(θ)rcos(θ)
Question 14
Question
Let f:R3→R2be a function defined by f(x, y, z)=(x2+yz, z2−xy). Find
the Jacobian matrix of fat the point (1,2,−1).
Solution
Step 1: Compute the partial derivatives of f:
∂f
∂x = (2x, −z),∂f
∂y = (z, x),∂f
∂z = (y, 2z)
Step 2: Evaluate the partial derivatives at the point (1,2,−1):
∂f
∂x (1,2,−1) = (2,1),∂f
∂y (1,2,−1) = (−1,1),∂f
∂z (1,2,−1) = (2,−2)
Step 3: Assemble the Jacobian matrix by arranging the partial derivatives:
Jf(1,2,−1) =
2 1
−1 1
2−2
11
Question 15
Question
Let f(x, y, z) =
ex+y
xyz
sin(z)
. Calculate the Jacobian matrix of fat the point
(0,1, π/2).
Solution
Step 1: Compute the partial derivatives of f:
∂f1
∂x =ex+y,∂f1
∂y =ex+y
∂f1
∂z = 0,∂f2
∂x =yz, ∂f2
∂y =xz
∂f2
∂z =xy, ∂f3
∂x = 0,∂f3
∂y = 0,∂f3
∂z = cos(z)
Step 2: Evaluate the partial derivatives at the point (0,1, π/2):
∂f1
∂x (0,1, π/2) = e0+1 =e
∂f1
∂y (0,1, π/2) = e0+1 =e
∂f1
∂z (0,1, π/2) = 0
∂f2
∂x (0,1, π/2) = 1 ·(π/2) = π/2
∂f2
∂y (0,1, π/2) = 0
∂f2
∂z (0,1, π/2) = 0
∂f3
∂x (0,1, π/2) = 0
∂f3
∂y (0,1, π/2) = 0
∂f3
∂z (0,1, π/2) = cos(π/2) = 0
Step 3: Assemble the Jacobian matrix of fat the point (0,1, π/2):
Jf(0,1, π/2) =
e e 0
π/200
0 0 0
12
Question 16
Question
Let f(x, y, z)=(x2y, yz, xz) be a vector-valued function. Find the Jacobian
matrix of f.
Solution
1. To find the Jacobian matrix of f, we first need to determine the partial
derivatives of each component function with respect to each variable.
Step 1: Compute ∂f1
∂x ,∂f1
∂y , and ∂f1
∂z for the first component function
f1(x, y, z) = x2y.
We have: ∂f1
∂x = 2xy, ∂f1
∂y =x2,∂f1
∂z = 0
Step 2: Compute ∂f2
∂x ,∂f2
∂y , and ∂f2
∂z for the second component function
f2(x, y, z) = yz.
We have: ∂f2
∂x = 0,∂f2
∂y =z, ∂f2
∂z =y
Step 3: Compute ∂f3
∂x ,∂f3
∂y , and ∂f3
∂z for the third component function
f3(x, y, z) = xz.
We have: ∂f3
∂x =z, ∂f3
∂y = 0,∂f3
∂z =x
2. Now, we can organize these partial derivatives into a 3×3 Jacobian matrix:
Jf=
∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z
∂f3
∂x
∂f3
∂y
∂f3
∂z
Plugging in the computed partial derivatives gives us the Jacobian matrix
for f:
Jf=
2xy x20
0z y
z0x
Therefore, the Jacobian matrix of f(x, y, z) = (x2y, yz, xz) is given by the
above expression.
13
Question 17
Question
Let f:R3→R2be a differentiable function given by f(x, y, z) = (x2+y, yz).
Find the Jacobian matrix of f.
Solution
Step 1: To find the Jacobian matrix of f, we need to compute the partial
derivatives of each component function.
Let f(x, y, z)=(u, v). Then u=x2+yand v=yz.
Step 2: Calculate the partial derivatives of uand vwith respect to x, y, and
z.
∂u
∂x = 2x, ∂u
∂y = 1,∂u
∂z = 0
∂v
∂x = 0,∂v
∂y =z, ∂v
∂z =y
Step 3: Assemble the partial derivatives into a 2 ×3 matrix known as the
Jacobian matrix of f.
Jf="∂u
∂x
∂u
∂y
∂u
∂z
∂v
∂x
∂v
∂y
∂v
∂z #=2x1 0
0z y
Therefore, the Jacobian matrix of fis 2x1 0
0z y.
Question 18
Question
Let f:R3→R2be defined by f(x, y, z) = (x2+y, yz). 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.
Step 1: Compute the partial derivatives of f:
∂
∂x(x2+y)=2x
∂
∂y (x2+y)=1
∂
∂z (xz) = y
14
Step 2: Assemble the partial derivatives into a matrix:
Jf="∂
∂x f1∂
∂y f1∂
∂z f1
∂
∂x f2∂
∂y f2∂
∂z f2#=2x1 0
0z y
Question 19
Question
Let f:R3→R3be defined by f(x, y, z)=(xy, exz + sin(y), x2z). Find the
Jacobian matrix of fat the point (1,0,1).
Solution
Step 1: Compute the partial derivatives of f.
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
where f1=xy,f2=exz + sin(y), and f3=x2z.
Calculating the partial derivatives:
∂f1
∂x =y, ∂f1
∂y =x, ∂f1
∂z = 0
∂f2
∂x =yz, ∂f2
∂y = cos(y),∂f2
∂z =xexz
∂f3
∂x = 2xz, ∂f3
∂y = 0,∂f3
∂z =x2
Step 2: Evaluate the Jacobian matrix at the point (1,0,1).
Substitute x= 1, y= 0, and z= 1 into the partial derivatives:
Jf(1,0,1) =
010
0 1 e
201
Question 20
Question
Let f:R3→R2be a function defined by f(x, y, z) = (x2+yz, xyz). Find the
Jacobian matrix of f.
15
Solution
To find the Jacobian matrix of f, we need to compute the partial derivatives of
each component function.
Step 1: Find ∂f1
∂x ,∂f1
∂y , and ∂f1
∂z .
∂f1
∂x = 2x∂f1
∂y =z∂f1
∂z =y
Step 2: Find ∂f2
∂x ,∂f2
∂y , and ∂f2
∂z .
∂f2
∂x =yz ∂f2
∂y =xz ∂f2
∂z =xy
Step 3: Assemble the Jacobian matrix Jf.
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 Jf=2x z y
yz xz xy.
Question 21
Question
Let f:R2→R3be a function defined by f(x, y)=(x2y, xy2, x +y). Determine
the Jacobian matrix of fat the point (1,2).
Solution
Step 1: Compute the partial derivatives of fwith respect to xand y:
∂f
∂x =
∂f1
∂x
∂f2
∂x
∂f3
∂x
=
2xy
y2
1
∂f
∂y =
∂f1
∂y
∂f2
∂y
∂f3
∂y
=
x2
2xy
1
Step 2: Evaluate the partial derivatives at the point (1,2):
∂f
∂x (1,2) =
4
4
1
16
∂f
∂y (1,2) =
1
4
1
Step 3: Construct the Jacobian matrix of fat the point (1,2):
Jacobian Matrix Jf(1,2) =
4 1
4 4
1 1
Question 22
Question
Let f:R3→R3be a function defined by f(x, y, z) = (2x2y+z, x2+ 3yz, y2).
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: Find ∂f1
∂x ,∂f1
∂y ,∂f1
∂z :
∂f1
∂x = 4xy, ∂f1
∂y = 2x2,∂f1
∂z = 1
Step 2: Find ∂f2
∂x ,∂f2
∂y ,∂f2
∂z :
∂f2
∂x = 2x, ∂f2
∂y = 3z, ∂f2
∂z = 3y
Step 3: Find ∂f3
∂x ,∂f3
∂y ,∂f3
∂z :
∂f3
∂x = 0,∂f3
∂y = 2y, ∂f3
∂z = 0
Therefore, the Jacobian matrix of fis:
Jf(x, y, z) =
4xy 2x21
2 3z3y
0 2y0
Question 23
Question
Let f(x, y, z) = x2y+yz3−3x. Find the Jacobian matrix of fat the point
(1,2,−1).
17
Solution
To find the Jacobian matrix of fat the point (1,2,−1), we need to compute
the partial derivatives of fwith respect to each of its variables at that point.
Step 1: Compute the partial derivative of fwith respect to x.The
partial derivative of fwith respect to xis given by:
∂f
∂x =∂
∂x(x2y+yz3−3x)=2xy −3
Step 2: Compute the partial derivative of fwith respect to y.The
partial derivative of fwith respect to yis given by:
∂f
∂y =∂
∂y (x2y+yz3−3x) = x2+z3
Step 3: Compute the partial derivative of fwith respect to z.The
partial derivative of fwith respect to zis given by:
∂f
∂z =∂
∂z (x2y+yz3−3x)=3yz2
Step 4: Evaluate the Jacobian matrix at the point (1,2,−1).Sub-
stitute x= 1, y= 2, and z=−1 into the partial derivatives calculated above:
∂f
∂x = 2(1)(2) −3 = 1 ∂f
∂y = (1)2+ (−1)3= 0 ∂f
∂z = 3(2)(−1)2=−6
Therefore, the Jacobian matrix of fat the point (1,2,−1) is:
Jf(1,2,−1) =
1
0
−6
Question 24
Question
Let f1(x, y) = x2+y2and f2(x, y) = xy −1. Find the Jacobian matrix of the
vector function f(x, y) = (f1(x, y), f2(x, y)) at the point (1,1).
Solution
Step 1: Calculate the partial derivatives of f1and f2with respect to xand y.
∂f1
∂x = 2x, ∂f1
∂y = 2y,
∂f2
∂x =y, ∂f2
∂y =x.
18
Step 2: Evaluate the partial derivatives at the point (1,1).
∂f1
∂x (1,1) = 2,∂f1
∂y (1,1) = 2,
∂f2
∂x (1,1) = 1,∂f2
∂y (1,1) = 1.
Step 3: Construct the Jacobian matrix of f(x, y) at the point (1,1).
Jf(1,1) = 2 2
1 1.
Question 25
Question
Let f:R3→R2be a function defined by f(x, y, z) = (x2+y2, y2+z2).
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 with respect to each input variable.
Step 1: Find ∂f1
∂x ,∂f1
∂y ,∂f1
∂z .
∂f1
∂x =∂
∂x(x2+y2)=2x
∂f1
∂y =∂
∂y (x2+y2)=2y
∂f1
∂z = 0
Step 2: Find ∂f2
∂x ,∂f2
∂y ,∂f2
∂z .
∂f2
∂x = 0
∂f2
∂y =∂
∂y (y2+z2) = 2y
∂f2
∂z =∂
∂z (y2+z2)=2z
Therefore, the Jacobian matrix of fis given by
Jf ="∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #=2x2y0
0 2y2z
19
Question 26
Question
Let f(x, y, z) =
x2y
yz
x+y+z
. Find the Jacobian matrix of fwith respect to
(x, y, z).
Solution
Step 1: Compute the partial derivatives of fwith respect to x,y, and z.
∂f
∂x =
2xy
0
1
∂f
∂y =
x2
z
1
∂f
∂z =
0
y
1
Step 2: Assemble the Jacobian matrix using the partial derivatives.
Jf=
∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z
∂f3
∂x
∂f3
∂y
∂f3
∂z
=
2xy x20
0z y
1 1 1
Therefore, the Jacobian matrix of fwith respect to (x, y, z) is
Jf=
2xy x20
0z y
1 1 1
Question 27
Question
Let f:R3→R2be a function defined by f(x, y, z) = (x2+y, y2−z). Determine
the Jacobian matrix of f.
Solution
To determine the Jacobian matrix of f, we need to find the partial derivatives
of each component function.
Step 1: Find ∂f1∂x,∂f1
∂y , and ∂f1
∂z .
∂f1
∂x =∂
∂x(x2+y)=2x
∂f1
∂y =∂
∂y (x2+y)=1
∂f1
∂z = 0
20
Step 2: Find ∂f2∂x,∂f2
∂y , and ∂f2
∂z .
∂f2
∂x = 0
∂f2
∂y =∂
∂y (y2−z)=2y
∂f2
∂z =−1
Step 3: Assemble the Jacobian matrix. The Jacobian matrix of fis
given by
"∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #=2x1 0
0 2y−1
Question 28
Question
Let f:R3→R2be a differentiable function defined by f(x, y, z) = (x2+y2, xz).
Find the Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to find the partial derivatives of each
component of fwith respect to each variable.
Step 1: Find the partial derivatives of fwith respect to x.
∂f1
∂x =∂(x2+y2)
∂x = 2x
∂f2
∂x =∂(xz)
∂x =z
Step 2: Find the partial derivatives of fwith respect to y.
∂f1
∂y =∂(x2+y2)
∂y = 2y
∂f2
∂y =∂(xz)
∂y = 0
Step 3: Find the partial derivatives of fwith respect to z.
∂f1
∂z =∂(x2+y2)
∂z = 0
∂f2
∂z =∂(xz)
∂z =x
21
Therefore, the Jacobian matrix of fis given by:
Jf(x, y, z) =
2x2y
z0
0x
Question 29
Question
Let f:R3→R2be a differentiable function defined by f(x, y, z)=(x2+yz, x −
z). Find the Jacobian matrix Jfof f.
Solution
To find the Jacobian matrix Jfof f, we need to compute the partial derivatives
of each component function of f.
Step 1: Compute ∂f1∂x,∂f1
∂y , and ∂f1
∂z .
∂f1
∂x =∂
∂x(x2+yz)=2x
∂f1
∂y =∂
∂y (x2+yz) = z
∂f1
∂z =∂
∂z (x2+yz) = y
Step 2: Compute ∂f2∂x,∂f2
∂y , and ∂f2
∂z .
∂f2
∂x =∂
∂x(x−z)=1
∂f2
∂y =∂
∂y (x−z)=0
∂f2
∂z =∂
∂z (x−z) = −1
Step 3: Construct the Jacobian matrix Jf.The Jacobian matrix Jf
of fis given by
Jf= ∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z !=2x z y
1 0 −1
Question 30
Question
Let f:R3→R2be defined by f(x, y, z) = (x2+y, yz). Find the Jacobian
matrix of fat the point (1,2,3).
22
Question 2
Question
Find the Jacobian matrix of the following system of equations:
(x=u2+v2
y=uv
Solution
Step 1: Compute the partial derivatives of xand ywith respect to uand v.
For x=u2+v2:∂x
∂u = 2uand ∂x
∂v = 2v
For y=uv:∂y
∂u =vand ∂y
∂v =u
Step 2: Assemble the partial derivatives into the Jacobian matrix. The
Jacobian matrix is given by:
J=∂x
∂u
∂x
∂v
∂y
∂u
∂y
∂v
Substitute the computed partial derivatives:
J=2u2v
v u
Therefore, the Jacobian matrix of the given system of equations is:
J=2u2v
v u
Question 3
Question
Let f:R3→R2be defined by f(x, y, z)=(x2yz, sin(yz)). Find the Jacobian
matrix of fat the point (1,0, π).
2
Solution
Step 1: Compute the partial derivatives of fwith respect to x,y, and z.
∂f1
∂x =∂
∂x(x2yz) = 2xyz,
∂f1
∂y =∂
∂y (x2yz) = x2z,
∂f1
∂z =∂
∂z (x2yz) = x2y,
∂f2
∂x =∂
∂x(sin(yz)) = 0,
∂f2
∂y =∂
∂y (sin(yz)) = zcos(yz),
∂f2
∂z =∂
∂z (sin(yz)) = ycos(yz).
Step 2: Evaluate the partial derivatives at the point (1,0, π).
∂f
∂x (1,0,π)
= 2(1)(0)(π) = 0,
∂f
∂y (1,0,π)
= (12)(π) = π,
∂f
∂z (1,0,π)
= (12)(0) = 0,
∂f
∂x (1,0,π)
= 0,
∂f
∂y (1,0,π)
= 0,
∂f
∂z (1,0,π)
= 0.
Step 3: Assemble the Jacobian matrix of fat the point (1,0, π).
Jacobian Matrix = 0π0
000.
Question 4
Question
Let f:R3→R3be a function defined by f(x, y, z) = (x2y+yz, xz, xyz).
Calculate the Jacobian matrix of f.
3
Solution
To find the Jacobian matrix of f, we need to calculate the partial derivatives of
each component function.
Step 1: Calculating the partial derivatives of fLet’s find the partial
derivatives of fwith respect to each variable.
For the first component function f1(x, y, z) = x2y+yz:
∂f1
∂x = 2xy, ∂f1
∂y =x2+z, ∂f1
∂z =y
For the second component function f2(x, y, z) = xz:
∂f2
∂x =z, ∂f2
∂y = 0,∂f2
∂z =x
For the third component function f3(x, y, z) = xyz:
∂f3
∂x =yz, ∂f3
∂y =xz, ∂f3
∂z =xy
Step 2: Constructing the Jacobian matrix The Jacobian matrix of f
is given by:
Jf=
∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z
∂f3
∂x
∂f3
∂y
∂f3
∂z
Substitute the partial derivatives we found into the Jacobian matrix:
Jf=
2xy x2+z y
z0x
yz xz xy
Therefore, the Jacobian matrix of fis:
Jf=
2xy x2+z y
z0x
yz xz xy
Question 5
Question
Let f:R2→R2be a mapping defined by f(x, y) = (x2+y2, xy). Compute the
Jacobian matrix of f.
4
Solution
To find the Jacobian matrix of f, we need to compute the partial derivatives of
each component function.
Step 1: Compute the partial derivative of the first component function with
respect to x.∂f1
∂x =∂
∂x(x2+y2)=2x
Step 2: Compute the partial derivative of the first component function with
respect to y.∂f1
∂y =∂
∂y (x2+y2)=2y
Step 3: Compute the partial derivative of the second component function
with respect to x.∂f2
∂x =∂
∂x(xy) = y
Step 4: Compute the partial derivative of the second component function
with respect to y.∂f2
∂y =∂
∂y (xy) = x
Step 5: Assemble the partial derivatives into the Jacobian matrix. The
Jacobian matrix of fis given by:
"∂f1
∂x
∂f1
∂y
∂f2
∂x
∂f2
∂y #=2x2y
y x
Question 6
Question
Let f(x, y)=(xy2, x2y) be a vector-valued function in R2. Determine the
Jacobian matrix of f(x, y).
Solution
To find the Jacobian matrix of f(x, y), we will compute the partial derivatives
of each component function.
Step 1: Find ∂f1
∂x and ∂f1
∂y .
∂f1
∂x =y2and ∂f1
∂y = 2xy
Step 2: Find ∂f2
∂x and ∂f2
∂y .
∂f2
∂x = 2xy and ∂f2
∂y =x2
5
Step 3: Assemble the Jacobian matrix. The Jacobian matrix of f(x, y) is
given by
Jf(x, y) = "∂f1
∂x
∂f1
∂y
∂f2
∂x
∂f2
∂y #
Substitute the partial derivatives we found into the matrix:
Jf(x, y) = y22xy
2xy x2
Therefore, the Jacobian matrix of f(x, y)=(xy2, x2y) is
Jf(x, y) = y22xy
2xy x2
Question 7
Question
Let f:R3→R2be a function defined by f(x, y, z) = (2x2y+z, xz2). Calculate
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:
Jf="∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #
Step 2: Let’s find the partial derivatives of fwith respect to x,y, and z.
∂f1
∂x = 4xy
∂f1
∂y = 2x2
∂f1
∂z = 1
Step 3: Similarly, ∂f2
∂x =z2
∂f2
∂y = 0
∂f2
∂z = 2xz
Step 4: Putting all the partial derivatives together, we get the Jacobian
matrix Jf:
Jf=4xy 2x21
z20 2xz
6
Question 8
Question
Let f:R3→R2be a function defined by f(x, y, z)=(x2+yz, xz). Find the
Jacobian matrix of f.
Solution
Step 1: The Jacobian matrix of a function f:Rn→Rmis an m×nmatrix
whose entries are the partial derivatives of the component functions of f. In this
case, the function f:R3→R2has component functions f1(x, y, z) = x2+yz
and f2(x, y, z) = xz.
Step 2: To find the Jacobian matrix of f, we first compute the partial
derivatives of f1and f2with respect to x,y, and z:
∂f1
∂x = 2x, ∂f1
∂y =z, ∂f1
∂z =y
∂f2
∂x =z, ∂f2
∂y = 0,∂f2
∂z =x
Step 3: Now, we construct the Jacobian matrix of fby arranging these
partial derivatives as rows:
Jf(x, y, z) = 2x z y
z0x
Therefore, the Jacobian matrix of fis given by
Jf(x, y, z) = 2x z y
z0x
Question 9
Question
Let f:R3→R2be a differentiable function defined by f(x, y, z)=(xy+yz, xz).
Find the Jacobian matrix of fat the point (1,2,3).
Solution
Step 1: Compute the partial derivatives of fwith respect to x,y, and z.
∂f1
∂x =∂
∂x(xy +yz) = y,
∂f1
∂y =∂
∂y (xy +yz) = x+z,
∂f1
∂z =∂
∂z (xy +yz) = y;
7
∂f2
∂x =∂
∂x(xz) = z,
∂f2
∂y =∂
∂y (xz)=0,
∂f2
∂z =∂
∂z (xz) = x.
Step 2: Evaluate the partial derivatives at the point (1,2,3).
∂f1
∂x (1,2,3) = 2,
∂f1
∂y (1,2,3) = 1 + 3 = 4,
∂f1
∂z (1,2,3) = 2;
∂f2
∂x (1,2,3) = 3,
∂f2
∂y (1,2,3) = 0,
∂f2
∂z (1,2,3) = 1.
Step 3: Assemble the Jacobian matrix of fat (1,2,3). The Jacobian matrix
of fat (1,2,3) is
Jf(1,2,3) = 242
301.
Question 10
Question
Let f(x, y, z) = x2y+yz3−sin(z). Find the Jacobian matrix of fat the point
(1,2, π).
Solution
To find the Jacobian matrix of fat the point (1,2, π), we need to compute the
partial derivatives of fwith respect to each variable x,y, and zat that point.
Step 1: Find ∂f ∂x
∂f
∂x =∂
∂x(x2y+yz3−sin(z)) = 2xy
Step 2: Find ∂f ∂y
∂f
∂y =∂
∂y (x2y+yz3−sin(z)) = x2+z3
8
Step 3: Find ∂f ∂z
∂f
∂z =∂
∂z (x2y+yz3−sin(z)) = 3yz2−cos(z)
Step 4: Evaluate the Jacobian matrix at (1,2, π) Now, substitute
x= 1, y= 2, and z=πinto the partial derivatives we found earlier:
∂f
∂x (1,2, π) = 2(1)(2) = 4
∂f
∂y (1,2, π)=12+ (π)3= 1 + π3
∂f
∂z (1,2, π) = 3(2)(π)2−cos(π)=6π2+ 1
Therefore, the Jacobian matrix of fat the point (1,2, π) is:
Jf(1,2, π) = 4 1 + π36π2+ 1
Question 11
Question
Let f(x, y, z)=(x2−yz, y3−xz, z2−xy). Find the Jacobian matrix of fwith
respect to x,y, and z.
Solution
Step 1: Compute the partial derivatives of fwith respect to x,y, and z.
∂f
∂x =
∂f1
∂x
∂f2
∂x
∂f3
∂x
=
2x−z−y
−z0y
−y−x2z
∂f
∂y =
∂f1
∂y
∂f2
∂y
∂f3
∂y
=
−z3y2−x
0 3y z
−x−z2y
∂f
∂z =
∂f1
∂z
∂f2
∂z
∂f3
∂z
=
−y0x
−x0z
2z−y−x
Step 2: Arrange the partial derivatives in a matrix to form the Jacobian
matrix of f.
Jf=
2x−z−y
−z0y
−y−x2z
,
−z3y2−x
0 3y z
−x−z2y
,
−y0x
−x0z
2z−y−x
9
Question 12
Question
Let f:R3→R2be a function defined by f(x, y, z)=(x2+yz, xz). 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.
Step 1: Compute ∂f1∂x,∂f1
∂y , and ∂f1
∂z
∂f1
∂x = 2x
∂f1
∂y =z
∂f1
∂z =y
Step 2: Compute ∂f2∂x,∂f2
∂y , and ∂f2
∂z
∂f2
∂x =z
∂f2
∂y = 0
∂f2
∂z =x
Step 3: Assemble the Jacobian matrix The Jacobian matrix of fis
given by
"∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #
Plugging in the computed partial derivatives, we have
2x z y
z0x
Question 13
Question
Find the Jacobian matrix of the transformation defined by x=rcos(θ) and
y=rsin(θ).
10
Solution
To find the Jacobian matrix of the transformation, we first need to express the
transformation in terms of rand θ.
Step 1: Express the transformation in terms of rand θ.
x=rcos(θ)
y=rsin(θ)
Step 2: Calculate the partial derivatives of the transformation with respect
to rand θ.∂x
∂r = cos(θ),∂x
∂θ =−rsin(θ)
∂y
∂r = sin(θ),∂y
∂θ =rcos(θ)
Step 3: Assemble the Jacobian matrix. The Jacobian matrix is given by:
J=∂x
∂r
∂x
∂θ
∂y
∂r
∂y
∂θ
Step 4: Substitute the partial derivatives into the Jacobian matrix.
J=cos(θ)−rsin(θ)
sin(θ)rcos(θ)
Therefore, the Jacobian matrix of the transformation is:
J=cos(θ)−rsin(θ)
sin(θ)rcos(θ)
Question 14
Question
Let f:R3→R2be a function defined by f(x, y, z)=(x2+yz, z2−xy). Find
the Jacobian matrix of fat the point (1,2,−1).
Solution
Step 1: Compute the partial derivatives of f:
∂f
∂x = (2x, −z),∂f
∂y = (z, x),∂f
∂z = (y, 2z)
Step 2: Evaluate the partial derivatives at the point (1,2,−1):
∂f
∂x (1,2,−1) = (2,1),∂f
∂y (1,2,−1) = (−1,1),∂f
∂z (1,2,−1) = (2,−2)
Step 3: Assemble the Jacobian matrix by arranging the partial derivatives:
Jf(1,2,−1) =
2 1
−1 1
2−2
11
Question 15
Question
Let f(x, y, z) =
ex+y
xyz
sin(z)
. Calculate the Jacobian matrix of fat the point
(0,1, π/2).
Solution
Step 1: Compute the partial derivatives of f:
∂f1
∂x =ex+y,∂f1
∂y =ex+y
∂f1
∂z = 0,∂f2
∂x =yz, ∂f2
∂y =xz
∂f2
∂z =xy, ∂f3
∂x = 0,∂f3
∂y = 0,∂f3
∂z = cos(z)
Step 2: Evaluate the partial derivatives at the point (0,1, π/2):
∂f1
∂x (0,1, π/2) = e0+1 =e
∂f1
∂y (0,1, π/2) = e0+1 =e
∂f1
∂z (0,1, π/2) = 0
∂f2
∂x (0,1, π/2) = 1 ·(π/2) = π/2
∂f2
∂y (0,1, π/2) = 0
∂f2
∂z (0,1, π/2) = 0
∂f3
∂x (0,1, π/2) = 0
∂f3
∂y (0,1, π/2) = 0
∂f3
∂z (0,1, π/2) = cos(π/2) = 0
Step 3: Assemble the Jacobian matrix of fat the point (0,1, π/2):
Jf(0,1, π/2) =
e e 0
π/200
0 0 0
12
Question 16
Question
Let f(x, y, z)=(x2y, yz, xz) be a vector-valued function. Find the Jacobian
matrix of f.
Solution
1. To find the Jacobian matrix of f, we first need to determine the partial
derivatives of each component function with respect to each variable.
Step 1: Compute ∂f1
∂x ,∂f1
∂y , and ∂f1
∂z for the first component function
f1(x, y, z) = x2y.
We have: ∂f1
∂x = 2xy, ∂f1
∂y =x2,∂f1
∂z = 0
Step 2: Compute ∂f2
∂x ,∂f2
∂y , and ∂f2
∂z for the second component function
f2(x, y, z) = yz.
We have: ∂f2
∂x = 0,∂f2
∂y =z, ∂f2
∂z =y
Step 3: Compute ∂f3
∂x ,∂f3
∂y , and ∂f3
∂z for the third component function
f3(x, y, z) = xz.
We have: ∂f3
∂x =z, ∂f3
∂y = 0,∂f3
∂z =x
2. Now, we can organize these partial derivatives into a 3×3 Jacobian matrix:
Jf=
∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z
∂f3
∂x
∂f3
∂y
∂f3
∂z
Plugging in the computed partial derivatives gives us the Jacobian matrix
for f:
Jf=
2xy x20
0z y
z0x
Therefore, the Jacobian matrix of f(x, y, z) = (x2y, yz, xz) is given by the
above expression.
13
Question 17
Question
Let f:R3→R2be a differentiable function given by f(x, y, z) = (x2+y, yz).
Find the Jacobian matrix of f.
Solution
Step 1: To find the Jacobian matrix of f, we need to compute the partial
derivatives of each component function.
Let f(x, y, z)=(u, v). Then u=x2+yand v=yz.
Step 2: Calculate the partial derivatives of uand vwith respect to x, y, and
z.
∂u
∂x = 2x, ∂u
∂y = 1,∂u
∂z = 0
∂v
∂x = 0,∂v
∂y =z, ∂v
∂z =y
Step 3: Assemble the partial derivatives into a 2 ×3 matrix known as the
Jacobian matrix of f.
Jf="∂u
∂x
∂u
∂y
∂u
∂z
∂v
∂x
∂v
∂y
∂v
∂z #=2x1 0
0z y
Therefore, the Jacobian matrix of fis 2x1 0
0z y.
Question 18
Question
Let f:R3→R2be defined by f(x, y, z) = (x2+y, yz). 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.
Step 1: Compute the partial derivatives of f:
∂
∂x(x2+y)=2x
∂
∂y (x2+y)=1
∂
∂z (xz) = y
14
Step 2: Assemble the partial derivatives into a matrix:
Jf="∂
∂x f1∂
∂y f1∂
∂z f1
∂
∂x f2∂
∂y f2∂
∂z f2#=2x1 0
0z y
Question 19
Question
Let f:R3→R3be defined by f(x, y, z)=(xy, exz + sin(y), x2z). Find the
Jacobian matrix of fat the point (1,0,1).
Solution
Step 1: Compute the partial derivatives of f.
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
where f1=xy,f2=exz + sin(y), and f3=x2z.
Calculating the partial derivatives:
∂f1
∂x =y, ∂f1
∂y =x, ∂f1
∂z = 0
∂f2
∂x =yz, ∂f2
∂y = cos(y),∂f2
∂z =xexz
∂f3
∂x = 2xz, ∂f3
∂y = 0,∂f3
∂z =x2
Step 2: Evaluate the Jacobian matrix at the point (1,0,1).
Substitute x= 1, y= 0, and z= 1 into the partial derivatives:
Jf(1,0,1) =
010
0 1 e
201
Question 20
Question
Let f:R3→R2be a function defined by f(x, y, z) = (x2+yz, xyz). Find the
Jacobian matrix of f.
15
Solution
To find the Jacobian matrix of f, we need to compute the partial derivatives of
each component function.
Step 1: Find ∂f1
∂x ,∂f1
∂y , and ∂f1
∂z .
∂f1
∂x = 2x∂f1
∂y =z∂f1
∂z =y
Step 2: Find ∂f2
∂x ,∂f2
∂y , and ∂f2
∂z .
∂f2
∂x =yz ∂f2
∂y =xz ∂f2
∂z =xy
Step 3: Assemble the Jacobian matrix Jf.
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 Jf=2x z y
yz xz xy.
Question 21
Question
Let f:R2→R3be a function defined by f(x, y)=(x2y, xy2, x +y). Determine
the Jacobian matrix of fat the point (1,2).
Solution
Step 1: Compute the partial derivatives of fwith respect to xand y:
∂f
∂x =
∂f1
∂x
∂f2
∂x
∂f3
∂x
=
2xy
y2
1
∂f
∂y =
∂f1
∂y
∂f2
∂y
∂f3
∂y
=
x2
2xy
1
Step 2: Evaluate the partial derivatives at the point (1,2):
∂f
∂x (1,2) =
4
4
1
16
∂f
∂y (1,2) =
1
4
1
Step 3: Construct the Jacobian matrix of fat the point (1,2):
Jacobian Matrix Jf(1,2) =
4 1
4 4
1 1
Question 22
Question
Let f:R3→R3be a function defined by f(x, y, z) = (2x2y+z, x2+ 3yz, y2).
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: Find ∂f1
∂x ,∂f1
∂y ,∂f1
∂z :
∂f1
∂x = 4xy, ∂f1
∂y = 2x2,∂f1
∂z = 1
Step 2: Find ∂f2
∂x ,∂f2
∂y ,∂f2
∂z :
∂f2
∂x = 2x, ∂f2
∂y = 3z, ∂f2
∂z = 3y
Step 3: Find ∂f3
∂x ,∂f3
∂y ,∂f3
∂z :
∂f3
∂x = 0,∂f3
∂y = 2y, ∂f3
∂z = 0
Therefore, the Jacobian matrix of fis:
Jf(x, y, z) =
4xy 2x21
2 3z3y
0 2y0
Question 23
Question
Let f(x, y, z) = x2y+yz3−3x. Find the Jacobian matrix of fat the point
(1,2,−1).
17
Solution
To find the Jacobian matrix of fat the point (1,2,−1), we need to compute
the partial derivatives of fwith respect to each of its variables at that point.
Step 1: Compute the partial derivative of fwith respect to x.The
partial derivative of fwith respect to xis given by:
∂f
∂x =∂
∂x(x2y+yz3−3x)=2xy −3
Step 2: Compute the partial derivative of fwith respect to y.The
partial derivative of fwith respect to yis given by:
∂f
∂y =∂
∂y (x2y+yz3−3x) = x2+z3
Step 3: Compute the partial derivative of fwith respect to z.The
partial derivative of fwith respect to zis given by:
∂f
∂z =∂
∂z (x2y+yz3−3x)=3yz2
Step 4: Evaluate the Jacobian matrix at the point (1,2,−1).Sub-
stitute x= 1, y= 2, and z=−1 into the partial derivatives calculated above:
∂f
∂x = 2(1)(2) −3 = 1 ∂f
∂y = (1)2+ (−1)3= 0 ∂f
∂z = 3(2)(−1)2=−6
Therefore, the Jacobian matrix of fat the point (1,2,−1) is:
Jf(1,2,−1) =
1
0
−6
Question 24
Question
Let f1(x, y) = x2+y2and f2(x, y) = xy −1. Find the Jacobian matrix of the
vector function f(x, y) = (f1(x, y), f2(x, y)) at the point (1,1).
Solution
Step 1: Calculate the partial derivatives of f1and f2with respect to xand y.
∂f1
∂x = 2x, ∂f1
∂y = 2y,
∂f2
∂x =y, ∂f2
∂y =x.
18
Step 2: Evaluate the partial derivatives at the point (1,1).
∂f1
∂x (1,1) = 2,∂f1
∂y (1,1) = 2,
∂f2
∂x (1,1) = 1,∂f2
∂y (1,1) = 1.
Step 3: Construct the Jacobian matrix of f(x, y) at the point (1,1).
Jf(1,1) = 2 2
1 1.
Question 25
Question
Let f:R3→R2be a function defined by f(x, y, z) = (x2+y2, y2+z2).
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 with respect to each input variable.
Step 1: Find ∂f1
∂x ,∂f1
∂y ,∂f1
∂z .
∂f1
∂x =∂
∂x(x2+y2)=2x
∂f1
∂y =∂
∂y (x2+y2)=2y
∂f1
∂z = 0
Step 2: Find ∂f2
∂x ,∂f2
∂y ,∂f2
∂z .
∂f2
∂x = 0
∂f2
∂y =∂
∂y (y2+z2) = 2y
∂f2
∂z =∂
∂z (y2+z2)=2z
Therefore, the Jacobian matrix of fis given by
Jf ="∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #=2x2y0
0 2y2z
19
Question 26
Question
Let f(x, y, z) =
x2y
yz
x+y+z
. Find the Jacobian matrix of fwith respect to
(x, y, z).
Solution
Step 1: Compute the partial derivatives of fwith respect to x,y, and z.
∂f
∂x =
2xy
0
1
∂f
∂y =
x2
z
1
∂f
∂z =
0
y
1
Step 2: Assemble the Jacobian matrix using the partial derivatives.
Jf=
∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z
∂f3
∂x
∂f3
∂y
∂f3
∂z
=
2xy x20
0z y
1 1 1
Therefore, the Jacobian matrix of fwith respect to (x, y, z) is
Jf=
2xy x20
0z y
1 1 1
Question 27
Question
Let f:R3→R2be a function defined by f(x, y, z) = (x2+y, y2−z). Determine
the Jacobian matrix of f.
Solution
To determine the Jacobian matrix of f, we need to find the partial derivatives
of each component function.
Step 1: Find ∂f1∂x,∂f1
∂y , and ∂f1
∂z .
∂f1
∂x =∂
∂x(x2+y)=2x
∂f1
∂y =∂
∂y (x2+y)=1
∂f1
∂z = 0
20
Step 2: Find ∂f2∂x,∂f2
∂y , and ∂f2
∂z .
∂f2
∂x = 0
∂f2
∂y =∂
∂y (y2−z)=2y
∂f2
∂z =−1
Step 3: Assemble the Jacobian matrix. The Jacobian matrix of fis
given by
"∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #=2x1 0
0 2y−1
Question 28
Question
Let f:R3→R2be a differentiable function defined by f(x, y, z) = (x2+y2, xz).
Find the Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to find the partial derivatives of each
component of fwith respect to each variable.
Step 1: Find the partial derivatives of fwith respect to x.
∂f1
∂x =∂(x2+y2)
∂x = 2x
∂f2
∂x =∂(xz)
∂x =z
Step 2: Find the partial derivatives of fwith respect to y.
∂f1
∂y =∂(x2+y2)
∂y = 2y
∂f2
∂y =∂(xz)
∂y = 0
Step 3: Find the partial derivatives of fwith respect to z.
∂f1
∂z =∂(x2+y2)
∂z = 0
∂f2
∂z =∂(xz)
∂z =x
21
Therefore, the Jacobian matrix of fis given by:
Jf(x, y, z) =
2x2y
z0
0x
Question 29
Question
Let f:R3→R2be a differentiable function defined by f(x, y, z)=(x2+yz, x −
z). Find the Jacobian matrix Jfof f.
Solution
To find the Jacobian matrix Jfof f, we need to compute the partial derivatives
of each component function of f.
Step 1: Compute ∂f1∂x,∂f1
∂y , and ∂f1
∂z .
∂f1
∂x =∂
∂x(x2+yz)=2x
∂f1
∂y =∂
∂y (x2+yz) = z
∂f1
∂z =∂
∂z (x2+yz) = y
Step 2: Compute ∂f2∂x,∂f2
∂y , and ∂f2
∂z .
∂f2
∂x =∂
∂x(x−z)=1
∂f2
∂y =∂
∂y (x−z)=0
∂f2
∂z =∂
∂z (x−z) = −1
Step 3: Construct the Jacobian matrix Jf.The Jacobian matrix Jf
of fis given by
Jf= ∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z !=2x z y
1 0 −1
Question 30
Question
Let f:R3→R2be defined by f(x, y, z) = (x2+y, yz). Find the Jacobian
matrix of fat the point (1,2,3).
22
Question 2
Question
Find the Jacobian matrix of the following system of equations:
(x=u2+v2
y=uv
Solution
Step 1: Compute the partial derivatives of xand ywith respect to uand v.
For x=u2+v2:∂x
∂u = 2uand ∂x
∂v = 2v
For y=uv:∂y
∂u =vand ∂y
∂v =u
Step 2: Assemble the partial derivatives into the Jacobian matrix. The
Jacobian matrix is given by:
J=∂x
∂u
∂x
∂v
∂y
∂u
∂y
∂v
Substitute the computed partial derivatives:
J=2u2v
v u
Therefore, the Jacobian matrix of the given system of equations is:
J=2u2v
v u
Question 3
Question
Let f:R3→R2be defined by f(x, y, z)=(x2yz, sin(yz)). Find the Jacobian
matrix of fat the point (1,0, π).
2
Solution
Step 1: Compute the partial derivatives of fwith respect to x,y, and z.
∂f1
∂x =∂
∂x(x2yz) = 2xyz,
∂f1
∂y =∂
∂y (x2yz) = x2z,
∂f1
∂z =∂
∂z (x2yz) = x2y,
∂f2
∂x =∂
∂x(sin(yz)) = 0,
∂f2
∂y =∂
∂y (sin(yz)) = zcos(yz),
∂f2
∂z =∂
∂z (sin(yz)) = ycos(yz).
Step 2: Evaluate the partial derivatives at the point (1,0, π).
∂f
∂x (1,0,π)
= 2(1)(0)(π) = 0,
∂f
∂y (1,0,π)
= (12)(π) = π,
∂f
∂z (1,0,π)
= (12)(0) = 0,
∂f
∂x (1,0,π)
= 0,
∂f
∂y (1,0,π)
= 0,
∂f
∂z (1,0,π)
= 0.
Step 3: Assemble the Jacobian matrix of fat the point (1,0, π).
Jacobian Matrix = 0π0
000.
Question 4
Question
Let f:R3→R3be a function defined by f(x, y, z) = (x2y+yz, xz, xyz).
Calculate the Jacobian matrix of f.
3
Solution
To find the Jacobian matrix of f, we need to calculate the partial derivatives of
each component function.
Step 1: Calculating the partial derivatives of fLet’s find the partial
derivatives of fwith respect to each variable.
For the first component function f1(x, y, z) = x2y+yz:
∂f1
∂x = 2xy, ∂f1
∂y =x2+z, ∂f1
∂z =y
For the second component function f2(x, y, z) = xz:
∂f2
∂x =z, ∂f2
∂y = 0,∂f2
∂z =x
For the third component function f3(x, y, z) = xyz:
∂f3
∂x =yz, ∂f3
∂y =xz, ∂f3
∂z =xy
Step 2: Constructing the Jacobian matrix The Jacobian matrix of f
is given by:
Jf=
∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z
∂f3
∂x
∂f3
∂y
∂f3
∂z
Substitute the partial derivatives we found into the Jacobian matrix:
Jf=
2xy x2+z y
z0x
yz xz xy
Therefore, the Jacobian matrix of fis:
Jf=
2xy x2+z y
z0x
yz xz xy
Question 5
Question
Let f:R2→R2be a mapping defined by f(x, y) = (x2+y2, xy). Compute the
Jacobian matrix of f.
4
Solution
To find the Jacobian matrix of f, we need to compute the partial derivatives of
each component function.
Step 1: Compute the partial derivative of the first component function with
respect to x.∂f1
∂x =∂
∂x(x2+y2)=2x
Step 2: Compute the partial derivative of the first component function with
respect to y.∂f1
∂y =∂
∂y (x2+y2)=2y
Step 3: Compute the partial derivative of the second component function
with respect to x.∂f2
∂x =∂
∂x(xy) = y
Step 4: Compute the partial derivative of the second component function
with respect to y.∂f2
∂y =∂
∂y (xy) = x
Step 5: Assemble the partial derivatives into the Jacobian matrix. The
Jacobian matrix of fis given by:
"∂f1
∂x
∂f1
∂y
∂f2
∂x
∂f2
∂y #=2x2y
y x
Question 6
Question
Let f(x, y)=(xy2, x2y) be a vector-valued function in R2. Determine the
Jacobian matrix of f(x, y).
Solution
To find the Jacobian matrix of f(x, y), we will compute the partial derivatives
of each component function.
Step 1: Find ∂f1
∂x and ∂f1
∂y .
∂f1
∂x =y2and ∂f1
∂y = 2xy
Step 2: Find ∂f2
∂x and ∂f2
∂y .
∂f2
∂x = 2xy and ∂f2
∂y =x2
5
Step 3: Assemble the Jacobian matrix. The Jacobian matrix of f(x, y) is
given by
Jf(x, y) = "∂f1
∂x
∂f1
∂y
∂f2
∂x
∂f2
∂y #
Substitute the partial derivatives we found into the matrix:
Jf(x, y) = y22xy
2xy x2
Therefore, the Jacobian matrix of f(x, y)=(xy2, x2y) is
Jf(x, y) = y22xy
2xy x2
Question 7
Question
Let f:R3→R2be a function defined by f(x, y, z) = (2x2y+z, xz2). Calculate
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:
Jf="∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #
Step 2: Let’s find the partial derivatives of fwith respect to x,y, and z.
∂f1
∂x = 4xy
∂f1
∂y = 2x2
∂f1
∂z = 1
Step 3: Similarly, ∂f2
∂x =z2
∂f2
∂y = 0
∂f2
∂z = 2xz
Step 4: Putting all the partial derivatives together, we get the Jacobian
matrix Jf:
Jf=4xy 2x21
z20 2xz
6
Question 8
Question
Let f:R3→R2be a function defined by f(x, y, z)=(x2+yz, xz). Find the
Jacobian matrix of f.
Solution
Step 1: The Jacobian matrix of a function f:Rn→Rmis an m×nmatrix
whose entries are the partial derivatives of the component functions of f. In this
case, the function f:R3→R2has component functions f1(x, y, z) = x2+yz
and f2(x, y, z) = xz.
Step 2: To find the Jacobian matrix of f, we first compute the partial
derivatives of f1and f2with respect to x,y, and z:
∂f1
∂x = 2x, ∂f1
∂y =z, ∂f1
∂z =y
∂f2
∂x =z, ∂f2
∂y = 0,∂f2
∂z =x
Step 3: Now, we construct the Jacobian matrix of fby arranging these
partial derivatives as rows:
Jf(x, y, z) = 2x z y
z0x
Therefore, the Jacobian matrix of fis given by
Jf(x, y, z) = 2x z y
z0x
Question 9
Question
Let f:R3→R2be a differentiable function defined by f(x, y, z)=(xy+yz, xz).
Find the Jacobian matrix of fat the point (1,2,3).
Solution
Step 1: Compute the partial derivatives of fwith respect to x,y, and z.
∂f1
∂x =∂
∂x(xy +yz) = y,
∂f1
∂y =∂
∂y (xy +yz) = x+z,
∂f1
∂z =∂
∂z (xy +yz) = y;
7
∂f2
∂x =∂
∂x(xz) = z,
∂f2
∂y =∂
∂y (xz)=0,
∂f2
∂z =∂
∂z (xz) = x.
Step 2: Evaluate the partial derivatives at the point (1,2,3).
∂f1
∂x (1,2,3) = 2,
∂f1
∂y (1,2,3) = 1 + 3 = 4,
∂f1
∂z (1,2,3) = 2;
∂f2
∂x (1,2,3) = 3,
∂f2
∂y (1,2,3) = 0,
∂f2
∂z (1,2,3) = 1.
Step 3: Assemble the Jacobian matrix of fat (1,2,3). The Jacobian matrix
of fat (1,2,3) is
Jf(1,2,3) = 242
301.
Question 10
Question
Let f(x, y, z) = x2y+yz3−sin(z). Find the Jacobian matrix of fat the point
(1,2, π).
Solution
To find the Jacobian matrix of fat the point (1,2, π), we need to compute the
partial derivatives of fwith respect to each variable x,y, and zat that point.
Step 1: Find ∂f ∂x
∂f
∂x =∂
∂x(x2y+yz3−sin(z)) = 2xy
Step 2: Find ∂f ∂y
∂f
∂y =∂
∂y (x2y+yz3−sin(z)) = x2+z3
8
Step 3: Find ∂f ∂z
∂f
∂z =∂
∂z (x2y+yz3−sin(z)) = 3yz2−cos(z)
Step 4: Evaluate the Jacobian matrix at (1,2, π) Now, substitute
x= 1, y= 2, and z=πinto the partial derivatives we found earlier:
∂f
∂x (1,2, π) = 2(1)(2) = 4
∂f
∂y (1,2, π)=12+ (π)3= 1 + π3
∂f
∂z (1,2, π) = 3(2)(π)2−cos(π)=6π2+ 1
Therefore, the Jacobian matrix of fat the point (1,2, π) is:
Jf(1,2, π) = 4 1 + π36π2+ 1
Question 11
Question
Let f(x, y, z)=(x2−yz, y3−xz, z2−xy). Find the Jacobian matrix of fwith
respect to x,y, and z.
Solution
Step 1: Compute the partial derivatives of fwith respect to x,y, and z.
∂f
∂x =
∂f1
∂x
∂f2
∂x
∂f3
∂x
=
2x−z−y
−z0y
−y−x2z
∂f
∂y =
∂f1
∂y
∂f2
∂y
∂f3
∂y
=
−z3y2−x
0 3y z
−x−z2y
∂f
∂z =
∂f1
∂z
∂f2
∂z
∂f3
∂z
=
−y0x
−x0z
2z−y−x
Step 2: Arrange the partial derivatives in a matrix to form the Jacobian
matrix of f.
Jf=
2x−z−y
−z0y
−y−x2z
,
−z3y2−x
0 3y z
−x−z2y
,
−y0x
−x0z
2z−y−x
9
Question 12
Question
Let f:R3→R2be a function defined by f(x, y, z)=(x2+yz, xz). 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.
Step 1: Compute ∂f1∂x,∂f1
∂y , and ∂f1
∂z
∂f1
∂x = 2x
∂f1
∂y =z
∂f1
∂z =y
Step 2: Compute ∂f2∂x,∂f2
∂y , and ∂f2
∂z
∂f2
∂x =z
∂f2
∂y = 0
∂f2
∂z =x
Step 3: Assemble the Jacobian matrix The Jacobian matrix of fis
given by
"∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #
Plugging in the computed partial derivatives, we have
2x z y
z0x
Question 13
Question
Find the Jacobian matrix of the transformation defined by x=rcos(θ) and
y=rsin(θ).
10
Solution
To find the Jacobian matrix of the transformation, we first need to express the
transformation in terms of rand θ.
Step 1: Express the transformation in terms of rand θ.
x=rcos(θ)
y=rsin(θ)
Step 2: Calculate the partial derivatives of the transformation with respect
to rand θ.∂x
∂r = cos(θ),∂x
∂θ =−rsin(θ)
∂y
∂r = sin(θ),∂y
∂θ =rcos(θ)
Step 3: Assemble the Jacobian matrix. The Jacobian matrix is given by:
J=∂x
∂r
∂x
∂θ
∂y
∂r
∂y
∂θ
Step 4: Substitute the partial derivatives into the Jacobian matrix.
J=cos(θ)−rsin(θ)
sin(θ)rcos(θ)
Therefore, the Jacobian matrix of the transformation is:
J=cos(θ)−rsin(θ)
sin(θ)rcos(θ)
Question 14
Question
Let f:R3→R2be a function defined by f(x, y, z)=(x2+yz, z2−xy). Find
the Jacobian matrix of fat the point (1,2,−1).
Solution
Step 1: Compute the partial derivatives of f:
∂f
∂x = (2x, −z),∂f
∂y = (z, x),∂f
∂z = (y, 2z)
Step 2: Evaluate the partial derivatives at the point (1,2,−1):
∂f
∂x (1,2,−1) = (2,1),∂f
∂y (1,2,−1) = (−1,1),∂f
∂z (1,2,−1) = (2,−2)
Step 3: Assemble the Jacobian matrix by arranging the partial derivatives:
Jf(1,2,−1) =
2 1
−1 1
2−2
11
Question 15
Question
Let f(x, y, z) =
ex+y
xyz
sin(z)
. Calculate the Jacobian matrix of fat the point
(0,1, π/2).
Solution
Step 1: Compute the partial derivatives of f:
∂f1
∂x =ex+y,∂f1
∂y =ex+y
∂f1
∂z = 0,∂f2
∂x =yz, ∂f2
∂y =xz
∂f2
∂z =xy, ∂f3
∂x = 0,∂f3
∂y = 0,∂f3
∂z = cos(z)
Step 2: Evaluate the partial derivatives at the point (0,1, π/2):
∂f1
∂x (0,1, π/2) = e0+1 =e
∂f1
∂y (0,1, π/2) = e0+1 =e
∂f1
∂z (0,1, π/2) = 0
∂f2
∂x (0,1, π/2) = 1 ·(π/2) = π/2
∂f2
∂y (0,1, π/2) = 0
∂f2
∂z (0,1, π/2) = 0
∂f3
∂x (0,1, π/2) = 0
∂f3
∂y (0,1, π/2) = 0
∂f3
∂z (0,1, π/2) = cos(π/2) = 0
Step 3: Assemble the Jacobian matrix of fat the point (0,1, π/2):
Jf(0,1, π/2) =
e e 0
π/200
0 0 0
12
Question 16
Question
Let f(x, y, z)=(x2y, yz, xz) be a vector-valued function. Find the Jacobian
matrix of f.
Solution
1. To find the Jacobian matrix of f, we first need to determine the partial
derivatives of each component function with respect to each variable.
Step 1: Compute ∂f1
∂x ,∂f1
∂y , and ∂f1
∂z for the first component function
f1(x, y, z) = x2y.
We have: ∂f1
∂x = 2xy, ∂f1
∂y =x2,∂f1
∂z = 0
Step 2: Compute ∂f2
∂x ,∂f2
∂y , and ∂f2
∂z for the second component function
f2(x, y, z) = yz.
We have: ∂f2
∂x = 0,∂f2
∂y =z, ∂f2
∂z =y
Step 3: Compute ∂f3
∂x ,∂f3
∂y , and ∂f3
∂z for the third component function
f3(x, y, z) = xz.
We have: ∂f3
∂x =z, ∂f3
∂y = 0,∂f3
∂z =x
2. Now, we can organize these partial derivatives into a 3×3 Jacobian matrix:
Jf=
∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z
∂f3
∂x
∂f3
∂y
∂f3
∂z
Plugging in the computed partial derivatives gives us the Jacobian matrix
for f:
Jf=
2xy x20
0z y
z0x
Therefore, the Jacobian matrix of f(x, y, z) = (x2y, yz, xz) is given by the
above expression.
13
Question 17
Question
Let f:R3→R2be a differentiable function given by f(x, y, z) = (x2+y, yz).
Find the Jacobian matrix of f.
Solution
Step 1: To find the Jacobian matrix of f, we need to compute the partial
derivatives of each component function.
Let f(x, y, z)=(u, v). Then u=x2+yand v=yz.
Step 2: Calculate the partial derivatives of uand vwith respect to x, y, and
z.
∂u
∂x = 2x, ∂u
∂y = 1,∂u
∂z = 0
∂v
∂x = 0,∂v
∂y =z, ∂v
∂z =y
Step 3: Assemble the partial derivatives into a 2 ×3 matrix known as the
Jacobian matrix of f.
Jf="∂u
∂x
∂u
∂y
∂u
∂z
∂v
∂x
∂v
∂y
∂v
∂z #=2x1 0
0z y
Therefore, the Jacobian matrix of fis 2x1 0
0z y.
Question 18
Question
Let f:R3→R2be defined by f(x, y, z) = (x2+y, yz). 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.
Step 1: Compute the partial derivatives of f:
∂
∂x(x2+y)=2x
∂
∂y (x2+y)=1
∂
∂z (xz) = y
14
Step 2: Assemble the partial derivatives into a matrix:
Jf="∂
∂x f1∂
∂y f1∂
∂z f1
∂
∂x f2∂
∂y f2∂
∂z f2#=2x1 0
0z y
Question 19
Question
Let f:R3→R3be defined by f(x, y, z)=(xy, exz + sin(y), x2z). Find the
Jacobian matrix of fat the point (1,0,1).
Solution
Step 1: Compute the partial derivatives of f.
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
where f1=xy,f2=exz + sin(y), and f3=x2z.
Calculating the partial derivatives:
∂f1
∂x =y, ∂f1
∂y =x, ∂f1
∂z = 0
∂f2
∂x =yz, ∂f2
∂y = cos(y),∂f2
∂z =xexz
∂f3
∂x = 2xz, ∂f3
∂y = 0,∂f3
∂z =x2
Step 2: Evaluate the Jacobian matrix at the point (1,0,1).
Substitute x= 1, y= 0, and z= 1 into the partial derivatives:
Jf(1,0,1) =
010
0 1 e
201
Question 20
Question
Let f:R3→R2be a function defined by f(x, y, z) = (x2+yz, xyz). Find the
Jacobian matrix of f.
15
Solution
To find the Jacobian matrix of f, we need to compute the partial derivatives of
each component function.
Step 1: Find ∂f1
∂x ,∂f1
∂y , and ∂f1
∂z .
∂f1
∂x = 2x∂f1
∂y =z∂f1
∂z =y
Step 2: Find ∂f2
∂x ,∂f2
∂y , and ∂f2
∂z .
∂f2
∂x =yz ∂f2
∂y =xz ∂f2
∂z =xy
Step 3: Assemble the Jacobian matrix Jf.
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 Jf=2x z y
yz xz xy.
Question 21
Question
Let f:R2→R3be a function defined by f(x, y)=(x2y, xy2, x +y). Determine
the Jacobian matrix of fat the point (1,2).
Solution
Step 1: Compute the partial derivatives of fwith respect to xand y:
∂f
∂x =
∂f1
∂x
∂f2
∂x
∂f3
∂x
=
2xy
y2
1
∂f
∂y =
∂f1
∂y
∂f2
∂y
∂f3
∂y
=
x2
2xy
1
Step 2: Evaluate the partial derivatives at the point (1,2):
∂f
∂x (1,2) =
4
4
1
16
∂f
∂y (1,2) =
1
4
1
Step 3: Construct the Jacobian matrix of fat the point (1,2):
Jacobian Matrix Jf(1,2) =
4 1
4 4
1 1
Question 22
Question
Let f:R3→R3be a function defined by f(x, y, z) = (2x2y+z, x2+ 3yz, y2).
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: Find ∂f1
∂x ,∂f1
∂y ,∂f1
∂z :
∂f1
∂x = 4xy, ∂f1
∂y = 2x2,∂f1
∂z = 1
Step 2: Find ∂f2
∂x ,∂f2
∂y ,∂f2
∂z :
∂f2
∂x = 2x, ∂f2
∂y = 3z, ∂f2
∂z = 3y
Step 3: Find ∂f3
∂x ,∂f3
∂y ,∂f3
∂z :
∂f3
∂x = 0,∂f3
∂y = 2y, ∂f3
∂z = 0
Therefore, the Jacobian matrix of fis:
Jf(x, y, z) =
4xy 2x21
2 3z3y
0 2y0
Question 23
Question
Let f(x, y, z) = x2y+yz3−3x. Find the Jacobian matrix of fat the point
(1,2,−1).
17
Solution
To find the Jacobian matrix of fat the point (1,2,−1), we need to compute
the partial derivatives of fwith respect to each of its variables at that point.
Step 1: Compute the partial derivative of fwith respect to x.The
partial derivative of fwith respect to xis given by:
∂f
∂x =∂
∂x(x2y+yz3−3x)=2xy −3
Step 2: Compute the partial derivative of fwith respect to y.The
partial derivative of fwith respect to yis given by:
∂f
∂y =∂
∂y (x2y+yz3−3x) = x2+z3
Step 3: Compute the partial derivative of fwith respect to z.The
partial derivative of fwith respect to zis given by:
∂f
∂z =∂
∂z (x2y+yz3−3x)=3yz2
Step 4: Evaluate the Jacobian matrix at the point (1,2,−1).Sub-
stitute x= 1, y= 2, and z=−1 into the partial derivatives calculated above:
∂f
∂x = 2(1)(2) −3 = 1 ∂f
∂y = (1)2+ (−1)3= 0 ∂f
∂z = 3(2)(−1)2=−6
Therefore, the Jacobian matrix of fat the point (1,2,−1) is:
Jf(1,2,−1) =
1
0
−6
Question 24
Question
Let f1(x, y) = x2+y2and f2(x, y) = xy −1. Find the Jacobian matrix of the
vector function f(x, y) = (f1(x, y), f2(x, y)) at the point (1,1).
Solution
Step 1: Calculate the partial derivatives of f1and f2with respect to xand y.
∂f1
∂x = 2x, ∂f1
∂y = 2y,
∂f2
∂x =y, ∂f2
∂y =x.
18
Step 2: Evaluate the partial derivatives at the point (1,1).
∂f1
∂x (1,1) = 2,∂f1
∂y (1,1) = 2,
∂f2
∂x (1,1) = 1,∂f2
∂y (1,1) = 1.
Step 3: Construct the Jacobian matrix of f(x, y) at the point (1,1).
Jf(1,1) = 2 2
1 1.
Question 25
Question
Let f:R3→R2be a function defined by f(x, y, z) = (x2+y2, y2+z2).
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 with respect to each input variable.
Step 1: Find ∂f1
∂x ,∂f1
∂y ,∂f1
∂z .
∂f1
∂x =∂
∂x(x2+y2)=2x
∂f1
∂y =∂
∂y (x2+y2)=2y
∂f1
∂z = 0
Step 2: Find ∂f2
∂x ,∂f2
∂y ,∂f2
∂z .
∂f2
∂x = 0
∂f2
∂y =∂
∂y (y2+z2) = 2y
∂f2
∂z =∂
∂z (y2+z2)=2z
Therefore, the Jacobian matrix of fis given by
Jf ="∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #=2x2y0
0 2y2z
19
Question 26
Question
Let f(x, y, z) =
x2y
yz
x+y+z
. Find the Jacobian matrix of fwith respect to
(x, y, z).
Solution
Step 1: Compute the partial derivatives of fwith respect to x,y, and z.
∂f
∂x =
2xy
0
1
∂f
∂y =
x2
z
1
∂f
∂z =
0
y
1
Step 2: Assemble the Jacobian matrix using the partial derivatives.
Jf=
∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z
∂f3
∂x
∂f3
∂y
∂f3
∂z
=
2xy x20
0z y
1 1 1
Therefore, the Jacobian matrix of fwith respect to (x, y, z) is
Jf=
2xy x20
0z y
1 1 1
Question 27
Question
Let f:R3→R2be a function defined by f(x, y, z) = (x2+y, y2−z). Determine
the Jacobian matrix of f.
Solution
To determine the Jacobian matrix of f, we need to find the partial derivatives
of each component function.
Step 1: Find ∂f1∂x,∂f1
∂y , and ∂f1
∂z .
∂f1
∂x =∂
∂x(x2+y)=2x
∂f1
∂y =∂
∂y (x2+y)=1
∂f1
∂z = 0
20
Step 2: Find ∂f2∂x,∂f2
∂y , and ∂f2
∂z .
∂f2
∂x = 0
∂f2
∂y =∂
∂y (y2−z)=2y
∂f2
∂z =−1
Step 3: Assemble the Jacobian matrix. The Jacobian matrix of fis
given by
"∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #=2x1 0
0 2y−1
Question 28
Question
Let f:R3→R2be a differentiable function defined by f(x, y, z) = (x2+y2, xz).
Find the Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to find the partial derivatives of each
component of fwith respect to each variable.
Step 1: Find the partial derivatives of fwith respect to x.
∂f1
∂x =∂(x2+y2)
∂x = 2x
∂f2
∂x =∂(xz)
∂x =z
Step 2: Find the partial derivatives of fwith respect to y.
∂f1
∂y =∂(x2+y2)
∂y = 2y
∂f2
∂y =∂(xz)
∂y = 0
Step 3: Find the partial derivatives of fwith respect to z.
∂f1
∂z =∂(x2+y2)
∂z = 0
∂f2
∂z =∂(xz)
∂z =x
21
Therefore, the Jacobian matrix of fis given by:
Jf(x, y, z) =
2x2y
z0
0x
Question 29
Question
Let f:R3→R2be a differentiable function defined by f(x, y, z)=(x2+yz, x −
z). Find the Jacobian matrix Jfof f.
Solution
To find the Jacobian matrix Jfof f, we need to compute the partial derivatives
of each component function of f.
Step 1: Compute ∂f1∂x,∂f1
∂y , and ∂f1
∂z .
∂f1
∂x =∂
∂x(x2+yz)=2x
∂f1
∂y =∂
∂y (x2+yz) = z
∂f1
∂z =∂
∂z (x2+yz) = y
Step 2: Compute ∂f2∂x,∂f2
∂y , and ∂f2
∂z .
∂f2
∂x =∂
∂x(x−z)=1
∂f2
∂y =∂
∂y (x−z)=0
∂f2
∂z =∂
∂z (x−z) = −1
Step 3: Construct the Jacobian matrix Jf.The Jacobian matrix Jf
of fis given by
Jf= ∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z !=2x z y
1 0 −1
Question 30
Question
Let f:R3→R2be defined by f(x, y, z) = (x2+y, yz). Find the Jacobian
matrix of fat the point (1,2,3).
22
Solution
Step 1: Compute the partial derivatives of fwith respect to x,y, and z:
∂f1
∂x = 2x, ∂f1
∂y = 1,∂f1
∂z = 0
∂f2
∂x = 0,∂f2
∂y =z, ∂f2
∂z =y
Step 2: Evaluate the partial derivatives at the point (1,2,3):
∂f1
∂x (1,2,3) = 2(1) = 2,∂f1
∂y (1,2,3) = 1,∂f1
∂z (1,2,3) = 0
∂f2
∂x (1,2,3) = 0,∂f2
∂y (1,2,3) = 3,∂f2
∂z (1,2,3) = 2
Step 3: Assemble the Jacobian matrix of fat (1,2,3):
Jf(1,2,3) = 210
032
23
