MATH 117 - ELEMENTS OF
MATHEMATICS - Jacobian matrix
Question Bank - Set 3
Liberty University
Question 1
Question
Let f:R3→R2be a function defined by f(x, y, z) = x2y, sin(z). Compute
the Jacobian matrix Jfof f.
Solution
Step 1: Compute the partial derivatives of fwith respect to x,y, and z.
∂f1
∂x = 2xy, ∂f2
∂x = 0
∂f1
∂y =x2,∂f2
∂y = 0
∂f1
∂z = 0,∂f2
∂z = cos(z)
Step 2: Assemble the Jacobian matrix Jfusing the partial derivatives com-
puted in step 1.
Jf="∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #
Jf=2xy x20
0 0 cos(z)
Question 2
Question
Let f(x, y) = x2+ 2y
3x−y2. Find the Jacobian matrix of fat the point (1,−1).
Solution
Step 1: Find the partial derivatives of fwith respect to xand y.
∂f
∂x =2x
3and ∂f
∂y =2
−2y
Step 2: Evaluate the partial derivatives at the point (1,−1).
∂f
∂x (1,−1) = 2
3and ∂f
∂y (1,−1) = 2
2
Step 3: Construct the Jacobian matrix of fat (1,−1).
Jf(1,−1) = 2 2
3−2
Therefore, the Jacobian matrix of fat the point (1,−1) is 2 2
3−2.
Question 3
Question
Let f(x, y, z) = (2xz −y2, x2+ 3yz, xy +z2). Find the Jacobian matrix of fat
the point (1,2,3).
Solution
Step 1: Compute the partial derivatives of f.
The Jacobian matrix of fat a point (x0, y0, z0) is given by
Jf(x0, y0, z0) =
∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z
∂f3
∂x
∂f3
∂y
∂f3
∂z
Here, f1= 2xz −y2,f2=x2+ 3yz, and f3=xy +z2.
Calculating the partial derivatives:
∂f1
∂x = 2z, ∂f1
∂y =−2y, ∂f1
∂z = 2x
∂f2
∂x = 2x, ∂f2
∂y = 3z, ∂f2
∂z = 3y
∂f3
∂x =y, ∂f3
∂y =x, ∂f3
∂z = 2z
Step 2: Evaluate the Jacobian matrix at the point (1,2,3).
2
Substitute x= 1, y= 2, and z= 3 into the partial derivatives above:
∂f1
∂x (1,2,3) = 6,∂f1
∂y (1,2,3) = −4,∂f1
∂z (1,2,3) = 2
∂f2
∂x (1,2,3) = 2,∂f2
∂y (1,2,3) = 9,∂f2
∂z (1,2,3) = 6
∂f3
∂x (1,2,3) = 2,∂f3
∂y (1,2,3) = 1,∂f3
∂z (1,2,3) = 6
Therefore, the Jacobian matrix of fat (1,2,3) is:
Jf(1,2,3) =
6−4 2
296
216
Question 4
Question
Let f:R3→R2be a function defined by f(x, y, z)=(x2y, yz). Find the
Jacobian matrix of fat the point (1,2,3).
Solution
To find the Jacobian matrix of fat the point (1,2,3), we need to compute the
partial derivatives of each component function at that point.
Step 1: Compute the partial derivatives. Let f1(x, y, z) = x2yand f2(x, y, z) =
yz. Then, the partial derivatives of fare:
∂f1
∂x = 2xy, ∂f1
∂y =x2,∂f1
∂z = 0
∂f2
∂x = 0,∂f2
∂y =z, ∂f2
∂z =y
Step 2: Evaluate the partial derivatives at the point (1,2,3). At (1,2,3),
we have:
∂f1
∂x (1,2,3) = 4,∂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. The Jacobian matrix of fat (1,2,3)
is given by:
Jf(1,2,3) = "∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #(1,2,3)
=410
032
3
Question 5
Question
Let f:R3→R2be a function defined by f(x, y, z)=(x2y, yz). Determine 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 variable.
Step 1: Calculate the partial derivatives of fLet’s find the partial
derivatives of f:
∂f1
∂x ,∂f1
∂y ,∂f1
∂z ,∂f2
∂x ,∂f2
∂y ,∂f2
∂z
Step 2: Calculate the partial derivatives
∂f1
∂x =∂
∂x (x2y)=2xy
∂f1
∂y =∂
∂y (x2y) = x2
∂f1
∂z = 0
∂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 !=2xy x20
0z y
Therefore, the Jacobian matrix of fis
Jf=2xy x20
0z y
Question 6
Question
Let f:R3→R2be the function defined by f(x, y, z)=(x2−y, 3z). Find 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: Find ∂f1
∂x ,∂f1
∂y , and ∂f1
∂z .
∂f1
∂x =∂
∂x (x2−y)=2x
∂f1
∂y =∂
∂y (x2−y) = −1
∂f1
∂z = 0
Step 2: Find ∂f2
∂x ,∂f2
∂y , and ∂f2
∂z .
∂f2
∂x =∂
∂x (3z) = 0
∂f2
∂y =∂
∂y (3z) = 0
∂f2
∂z =∂
∂z (3z)=3
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 !=2x−1 0
0 0 3
Question 7
Question
Let f:R2→R2be a function defined by f(x, y) = (x2+y2, xy). Determine
the Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we compute the partial derivatives of each
component function of fwith respect to xand y.
Step 1: Compute ∂f1
∂x and ∂f1
∂y for the first component function f1(x, y) =
x2+y2.
∂f1
∂x = 2xand ∂f1
∂y = 2y
Step 2: Compute ∂f2
∂x and ∂f2
∂y for the second component function f2(x, y) =
xy.
∂f2
∂x =yand ∂f2
∂y =x
5
Step 3: Assemble the partial derivatives into a 2 ×2 matrix to obtain the
Jacobian matrix of f.
Jacobian matrix of f="∂f1
∂x
∂f1
∂y
∂f2
∂x
∂f2
∂y #=2x2y
y x
Question 8
Question
Let f:R3→R2be a vector function defined by f(x, y, z)=(x2+y, xy +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.
Step 1: Compute ∂f1
∂x ,∂f1
∂y , and ∂f1
∂z .
∂f1
∂x = 2x
∂f1
∂y = 1
∂f1
∂z = 0
Step 2: Compute ∂f2
∂x ,∂f2
∂y , and ∂f2
∂z .
∂f2
∂x =y
∂f2
∂y =x
∂f2
∂z = 1
Step 3: Assemble the partial derivatives into the Jacobian matrix. The
Jacobian matrix of fis given by
Jf="∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #=2x1 0
y x 1
Question 9
Question
Let f:R3→R2be a function defined by f(x, y, z) = (x2y, yz). Compute the
Jacobian matrix of f.
6
Solution
To find the Jacobian matrix of f, we need to compute the partial derivatives of
fwith respect to each variable and arrange them in a matrix.
Step 1: Compute ∂f1
∂x ,∂f1
∂y , and ∂f1
∂z for the first component of f.
Given f(x, y, z) = (x2y, yz), we have:
f1(x, y, z) = x2y
Calculating the partial derivatives:
∂f1
∂x = 2xy, ∂f1
∂y =x2,∂f1
∂z = 0
Step 2: Compute ∂f2
∂x ,∂f2
∂y , and ∂f2
∂z for the second component of f.
Given f(x, y, z) = (x2y, yz), we have:
f2(x, y, z) = yz
Calculating the partial derivatives:
∂f2
∂x = 0,∂f2
∂y =z, ∂f2
∂z =y
Step 3: Form the Jacobian matrix Jfusing the computed partial deriva-
tives.
The Jacobian matrix Jfof fis given by:
Jf="∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #
Substitute the partial derivatives into the matrix:
Jf=2xy x20
0z y
Therefore, the Jacobian matrix of fis:
Jf=2xy x20
0z y
Question 10
Question
Let f:R3→R3be a transformation defined by f(x, y, z)=(x2+y, y2+z, z2+
x). Find the Jacobian matrix of f.
7
Solution
Step 1: Compute the partial derivatives of fwith respect to x, y, z.
∂f1
∂x = 2x, ∂f1
∂y = 1,∂f1
∂z = 0,
∂f2
∂x = 0,∂f2
∂y = 2y, ∂f2
∂z = 1,
∂f3
∂x = 1,∂f3
∂y = 0,∂f3
∂z = 2z.
Step 2: Assemble the partial derivatives into a 3 ×3 Jacobian matrix.
Jf(x, y, z) =
∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z
∂f3
∂x
∂f3
∂y
∂f3
∂z
=
2x1 0
0 2y1
1 0 2z
.
Therefore, the Jacobian matrix of fis
2x1 0
0 2y1
1 0 2z
.
Question 11
Question
Let f(x, y, z)=(x2+yz, y2+xz, z2+xy) be a vector-valued function. Find the
Jacobian matrix of f.
Solution
Step 1: Compute the partial derivatives of fwith respect to x,y, and z.
∂f
∂x =
∂
∂x (x2+yz)
∂
∂x (y2+xz)
∂
∂x (z2+xy)
=
2x
z
y
Step 2: Compute the partial derivatives of fwith respect to y.
∂f
∂y =
∂
∂y (x2+yz)
∂
∂y (y2+xz)
∂
∂y (z2+xy)
=
z
2y
x
Step 3: Compute the partial derivatives of fwith respect to z.
∂f
∂z =
∂
∂z (x2+yz)
∂
∂z (y2+xz)
∂
∂z (z2+xy)
=
y
x
2z
8
Step 4: Assemble the partial derivatives into the Jacobian matrix of f.
Jf=
2x z y
z2y x
y x 2z
Question 12
Question
Let f:R2→R2be defined by f(x, y) = (3x2+ 2y, x3+y2). Calculate the
Jacobian matrix of fat the point (1,1).
Solution
Step 1: Compute the partial derivatives of fwith respect to xand y.
∂f1
∂x = 6x, ∂f1
∂y = 2
∂f2
∂x = 3x2,∂f2
∂y = 2y
Step 2: Evaluate the partial derivatives at the point (1,1).
∂f1
∂x (1,1) = 6(1) = 6,∂f1
∂y (1,1) = 2
∂f2
∂x (1,1) = 3(1)2= 3,∂f2
∂y (1,1) = 2(1) = 2
Step 3: Assemble the Jacobian matrix by arranging the partial derivatives.
Jf(1,1) = "∂f1
∂x (1,1) ∂f1
∂y (1,1)
∂f2
∂x (1,1) ∂f2
∂y (1,1)#=6 2
3 2
Therefore, the Jacobian matrix of fat the point (1,1) is 6 2
3 2.
Question 13
Question
Let f:R3→R2be a mapping defined by f(x, y, z) = (x2+yz, x +y+z).
Calculate the Jacobian matrix of f.
9
Solution
Step 1: Find the partial derivatives of fwith respect to each variable. The
Jacobian matrix of fis given by:
J="∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #
where f1(x, y, z) = x2+yz and f2(x, y, z) = x+y+z.
Step 2: Find the partial derivatives of f1:
∂f1
∂x = 2x, ∂f1
∂y =z, ∂f1
∂z =y
Step 3: Find the partial derivatives of f2:
∂f2
∂x = 1,∂f2
∂y = 1,∂f2
∂z = 1
Step 4: Assemble the Jacobian matrix Jusing the partial derivatives cal-
culated above:
J=2x z y
1 1 1
Question 14
Question
Let f:R3→R2be a function defined by f(x, y, z)=(x2+yz, y2+xz). Find
the Jacobian matrix of f.
Solution
Step 1: To find the Jacobian matrix of f, we first need to compute the partial
derivatives of each component function.
Step 2: The partial derivatives of fare:
∂f1
∂x = 2x, ∂f1
∂y =z, ∂f1
∂z =y
∂f2
∂x =z, ∂f2
∂y = 2y, ∂f2
∂z =x
Step 3: Using the partial derivatives, we can construct the Jacobian matrix
of f:
Jf="∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #
Jf=2x z y
z2y x
Therefore, the Jacobian matrix of fis 2x z y
z2y x.
10
Question 15
Question
Let f(x, y, z) = x2+ 2y2+ 3z2+ 4xy + 5xz + 6yz. Find the Jacobian matrix of
fat the point (1,1,1).
Solution
Step 1: Compute the partial derivatives of fwith respect to x,y, and z.
∂f
∂x = 2x+ 4y+ 5z
∂f
∂y = 4y+ 6z
∂f
∂z = 6z+ 5x
Step 2: Evaluate the partial derivatives at the point (1,1,1).
∂f
∂x (1,1,1) = 2(1) + 4(1) + 5(1) = 2 + 4 + 5 = 11
∂f
∂y (1,1,1) = 4(1) + 6(1) = 4 + 6 = 10
∂f
∂z (1,1,1) = 6(1) + 5(1) = 6 + 5 = 11
Step 3: Construct the Jacobian matrix of fat (1,1,1). The Jacobian matrix
is a 1 ×3 matrix where the entries are the partial derivatives evaluated at the
point.
Jacobian matrix of fat (1,1,1) : 11 10 11
Question 16
Question
Let f:R3→R2be a function defined by f(x, y, z)=(xy, yz). Determine the
Jacobian matrix of f.
Solution
Step 1: The Jacobian matrix of a function f:Rn→Rmis defined as the matrix
of all first-order partial derivatives of f. The Jacobian matrix of fwill be a
m×nmatrix.
Step 2: Given f(x, y, z)=(xy, yz), let’s find the partial derivatives of f:
∂f1
∂x =y, ∂f1
∂y =x, ∂f1
∂z = 0
11
∂f2
∂x = 0,∂f2
∂y =z, ∂f2
∂z =y
Step 3: The Jacobian matrix of fis:
J="∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #=y x 0
0z y
Question 17
Question
Let f(x, y, z) = xy2z3be a function in R3. Find the Jacobian matrix of fat
the point (1,2,3).
Solution
Step 1: Compute the partial derivatives of fwith respect to each variable.
∂f
∂x =y2z3
∂f
∂y = 2xyz3
∂f
∂z = 3xy2z2
Step 2: Evaluate the partial derivatives at the point (1,2,3).
∂f
∂x = 22·33= 36
∂f
∂y = 2 ·1·33= 54
∂f
∂z = 3 ·22·32= 108
Step 3: Assemble the Jacobian matrix using the partial derivatives. The
Jacobian matrix of fat (1,2,3) is given by:
Jf(1,2,3) = h∂f
∂x
∂f
∂y
∂f
∂z i=36 54 108
Question 18
Question
Let f:R3→R2be the function defined by f(x, y, z) = (x2y, yz). Compute the
Jacobian matrix of f.
12
Solution
Step 1: Compute the partial derivatives of fwith respect to x,y, and z.
∂f
∂x =2xy
0,∂f
∂y =x2
z,∂f
∂z =0
y
Step 2: Assemble the partial derivatives into the Jacobian matrix of f.
Jf= ∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z !=2xy x20
0z y
Question 19
Question
Let f:R3→R2be a function defined by f(x, y, z)=(xy+z, x2−yz). Compute
the Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to compute the partial derivatives of
fwith respect to each input variable x,y, and z.
Step 1: Compute ∂f
∂x
Taking the partial derivative of fwith respect to x, we get:
∂f
∂x =∂
∂x (xy +z)
∂
∂x (x2−yz)=y
2x
Step 2: Compute ∂f
∂y
Taking the partial derivative of fwith respect to y, we get:
∂f
∂y = ∂
∂y (xy +z)
∂
∂y (x2−yz)!=x
−x
Step 3: Compute ∂f
∂z
Taking the partial derivative of fwith respect to z, we get:
∂f
∂z =∂
∂z (xy +z)
∂
∂z (x2−yz)=1
−y
Step 4: Construct the Jacobian matrix J(f)
The Jacobian matrix of fis given by:
J(f) = ∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z !=y x 1
2x−x−y
Therefore, the Jacobian matrix of fis y x 1
2x−x−y.
13
Question 20
Question
Let f:R3→R3be a vector field defined by f(x, y, z) = (xy2, xz, x2y+z).
Determine the Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to compute the matrix of partial
derivatives of fwith respect to x,y, and z.
Step 1: Compute ∂f1∂x, ∂f1
∂y ,∂f1
∂z
∂f1
∂x =∂
∂x (xy2) = y2
∂f1
∂y =∂
∂y (xy2)=2xy
∂f1
∂z =∂
∂z (xy2)=0
Step 2: Compute ∂f2∂x, ∂f2
∂y ,∂f2
∂z
∂f2
∂x =∂
∂x (xz) = z
∂f2
∂y =∂
∂y (xz)=0
∂f2
∂z =∂
∂z (xz) = x
Step 3: Compute ∂f3∂x, ∂f3
∂y ,∂f3
∂z
∂f3
∂x =∂
∂x (x2y+z)=2xy
∂f3
∂y =∂
∂y (x2y+z) = x2
∂f3
∂z =∂
∂z (x2y+z)=1
Step 4: Assemble the Jacobian matrix The Jacobian matrix of fis
Jf=
y22xy 0
z0x
2xy x21
Question 21
Question
Let f:R3→R2be a function defined by f(x, y, z) = (x2+y, y2z). Compute
the Jacobian matrix of fat the point (1,2,3).
14
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 = 2yz, ∂f2
∂z =y2
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) = 2(3) = 6,∂f2
∂z (1,2,3) = 22= 4
Step 3: Assemble the Jacobian matrix of fat (1,2,3).
Jf(1,2,3) = ∂f1
∂x (1,2,3) ∂f1
∂y (1,2,3) ∂f1
∂z (1,2,3)
∂f2
∂x (1,2,3) ∂f2
∂y (1,2,3) ∂f2
∂z (1,2,3)!=210
064
Question 22
Question
Let f:R2→R2be a function defined by f(x, y) = (x2+y, y2−x). Find the
Jacobian matrix of fat the point (1,2).
Solution
Step 1: Compute the partial derivatives of f:
∂f1
∂x = 2x, ∂f1
∂y = 1,
∂f2
∂x =−1,∂f2
∂y = 2y.
Step 2: Evaluate the partial derivatives at the point (1,2):
∂f1
∂x (1,2) = 2,∂f1
∂y (1,2) = 1,
∂f2
∂x (1,2) = −1,∂f2
∂y (1,2) = 4.
Step 3: Construct the Jacobian matrix Jf(1,2):
Jf(1,2) = "∂f1
∂x (1,2) ∂f1
∂y (1,2)
∂f2
∂x (1,2) ∂f2
∂y (1,2)#=2 1
−1 4.
Therefore, the Jacobian matrix of fat (1,2) is 2 1
−1 4.
15
Question 23
Question
Let f(x, y) = x2y
2xy2. Compute the Jacobian matrix Jf(x, y) of fat the point
(1,2).
Solution
Step 1: Compute the partial derivatives of fwith respect to xand y.
∂f
∂x =2xy
2y2,∂f
∂y =x2
4xy
Step 2: Evaluate the partial derivatives at the point (1,2).
∂f
∂x (1,2) = 4
4,∂f
∂y (1,2) = 1
8
Step 3: Construct the Jacobian matrix Jf(1,2).
Jf(1,2) = ∂f1
∂x
∂f1
∂y
∂f2
∂x
∂f2
∂y !=4 1
4 8
Therefore, the Jacobian matrix of fat the point (1,2) is 4 1
4 8.
Question 24
Question
Let f(x, y, z) = xyz2and g(x, y, z) = x2y+y2z+z2x. Find the Jacobian matrix
of the function
h(x, y, z)=(f, g) at the point (1,2,−1).
Solution
Step 1: Compute the partial derivatives of fand gwith respect to x,y, and z.
For f(x, y, z) = xyz2:
∂f
∂x =yz2
∂f
∂y =xz2
∂f
∂z = 2xyz
16
For g(x, y, z) = x2y+y2z+z2x:
∂g
∂x = 2xy +z2
∂g
∂y =x2+ 2yz
∂g
∂z =y2+ 2zx
Step 2: Evaluate the partial derivatives at the point (1,2,−1).
For f:
∂f
∂x = 2
∂f
∂y =−2
∂f
∂z =−4
For g:
∂g
∂x = 4
∂g
∂y = 5
∂g
∂z = 1
Step 3: Assemble the Jacobian matrix of
hat (1,2,−1).
J
h(1,2,−1) = "∂f
∂x
∂f
∂y
∂f
∂z
∂g
∂x
∂g
∂y
∂g
∂z #=2−2−4
4 5 1
Question 25
Question
Let f(x, y, z) = xy2zand g(x, y, z) = x2+y2+z2. Determine the Jacobian
matrix of the composite function h(x, y, z) = f(g(x, y, z), x, y).
Solution
Step 1: Compute the partial derivatives of fwith respect to x,y, and z.
∂f
∂x =y2z, ∂f
∂y = 2xyz, ∂f
∂z =xy2
17
Step 2: Compute the partial derivatives of gwith respect to x,y, and z.
∂g
∂x = 2x, ∂g
∂y = 2y, ∂g
∂z = 2z
Step 3: Express hin terms of fand g.
h(x, y, z) = f(g(x, y, z), x, y) = f(g(x, y, z), x, y) = f(g, x, y) = gxy2
Step 4: Compute the partial derivatives of hwith respect to x,y, and z.
∂h
∂x =y2,∂h
∂y = 2xy, ∂h
∂z = 0
Step 5: Write the Jacobian matrix of h.
Jh=h∂h
∂x
∂h
∂y
∂h
∂z i=y22xy 0
Question 26
Question
Consider the following functions f1(x, y) = x2+ 2yand f2(x, y) = y3−x.
Calculate the Jacobian matrix of the mapping (f1, f2) : R2→R2at the point
(1,2).
Solution
Given functions f1(x, y) = x2+ 2yand f2(x, y) = y3−x, we are asked to
calculate the Jacobian matrix of the mapping (f1, f2) : R2→R2at the point
(1,2).
The Jacobian matrix of a mapping f:Rn→Rmis given by:
J(f) =
∂f1
∂x1
∂f1
∂x2
· · · ∂f1
∂xn
∂f2
∂x1
∂f2
∂x2
· · · ∂f2
∂xn
.
.
..
.
.....
.
.
∂fm
∂x1
∂fm
∂x2
· · · ∂fm
∂xn
In our case, n= 2 and m= 2. So we need to compute the partial derivatives
of f1and f2with respect to xand y.
Step 1: Compute partial derivatives of f1
∂f1
∂x = 2xand ∂f1
∂y = 2
Step 2: Compute partial derivatives of f2
∂f2
∂x =−1 and ∂f2
∂y = 3y2
18
Step 3: Evaluate the Jacobian matrix at (1,2) Substitute x= 1 and
y= 2 into the partial derivatives:
J(f) = 2 2
−1 12
So, the Jacobian matrix of the mapping (f1, f2) at the point (1,2) is:
J(f) = 2 2
−1 12
Question 27
Question
Let f:R3→R2be a mapping defined by f(x, y, z)=(x2+yz, y2+xz). Find
the Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we first need to compute the partial derivatives
of each component function.
Step 1: Find the partial derivatives of f: Let f(x, y, z)=(u, v).
∂u
∂x =∂
∂x (x2+yz)=2x
∂u
∂y =∂
∂y (x2+yz) = z
∂u
∂z =∂
∂z (x2+yz) = y
∂v
∂x =∂
∂x (y2+xz) = z
∂v
∂y =∂
∂y (y2+xz)=2y
∂v
∂z =∂
∂z (y2+xz) = x
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 into the matrix:
19
Jf=2x z y
z2y x
Therefore, the Jacobian matrix of fis
Jf=2x z y
z2y x
.
Question 28
Question
Let f(x) =
x2y2
2xy
z2
. Find the Jacobian matrix Jf(x) of f.
Solution
Step 1: The Jacobian matrix Jf(x) of f(x) is given by:
Jf(x) =
∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z
∂f3
∂x
∂f3
∂y
∂f3
∂z
Step 2: Compute the partial derivatives of f(x): Let f1=x2y2,f2= 2xy,
and f3=z2. We have:
∂f1
∂x = 2xy2,∂f1
∂y = 2x2y, ∂f1
∂z = 0
∂f2
∂x = 2y, ∂f2
∂y = 2x, ∂f2
∂z = 0
∂f3
∂x = 0,∂f3
∂y = 0,∂f3
∂z = 2z
Step 3: Assemble the partial derivatives into the Jacobian matrix:
Jf(x) =
2xy22x2y0
2y2x0
0 0 2z
Therefore, the Jacobian matrix of f(x) is:
Jf(x) =
2xy22x2y0
2y2x0
0 0 2z
20
Question 29
Question
Find the Jacobian matrix of the following transformation:
x=u2+v2
y= 2u−v3
Solution
To find the Jacobian matrix of a transformation, we need to find the partial
derivatives of each component function with respect to each input variable.
Step 1: Find ∂x
∂u ,∂x
∂v ,∂y
∂u ,∂y
∂v .
For x=u2+v2:∂x
∂u = 2u
∂x
∂v = 2v
For y= 2u−v3:
∂y
∂u = 2
∂y
∂v =−3v2
Step 2: Assemble the Jacobian matrix.
The Jacobian matrix is given by:
J=∂x
∂u
∂x
∂v
∂y
∂u
∂y
∂v
Plugging in the partial derivative values we found in Step 1:
J=2u2v
2−3v2
Therefore, the Jacobian matrix of the transformation is:
J=2u2v
2−3v2
Question 30
Question
Let f(x, y, z)=(x3+y3+z3, xyz). Compute the Jacobian matrix of f.
21
Solution
Step 1: Find the partial derivatives of fwith respect to xand y.
∂f
∂x =2x
3and ∂f
∂y =2
−2y
Step 2: Evaluate the partial derivatives at the point (1,−1).
∂f
∂x (1,−1) = 2
3and ∂f
∂y (1,−1) = 2
2
Step 3: Construct the Jacobian matrix of fat (1,−1).
Jf(1,−1) = 2 2
3−2
Therefore, the Jacobian matrix of fat the point (1,−1) is 2 2
3−2.
Question 3
Question
Let f(x, y, z) = (2xz −y2, x2+ 3yz, xy +z2). Find the Jacobian matrix of fat
the point (1,2,3).
Solution
Step 1: Compute the partial derivatives of f.
The Jacobian matrix of fat a point (x0, y0, z0) is given by
Jf(x0, y0, z0) =
∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z
∂f3
∂x
∂f3
∂y
∂f3
∂z
Here, f1= 2xz −y2,f2=x2+ 3yz, and f3=xy +z2.
Calculating the partial derivatives:
∂f1
∂x = 2z, ∂f1
∂y =−2y, ∂f1
∂z = 2x
∂f2
∂x = 2x, ∂f2
∂y = 3z, ∂f2
∂z = 3y
∂f3
∂x =y, ∂f3
∂y =x, ∂f3
∂z = 2z
Step 2: Evaluate the Jacobian matrix at the point (1,2,3).
2
Substitute x= 1, y= 2, and z= 3 into the partial derivatives above:
∂f1
∂x (1,2,3) = 6,∂f1
∂y (1,2,3) = −4,∂f1
∂z (1,2,3) = 2
∂f2
∂x (1,2,3) = 2,∂f2
∂y (1,2,3) = 9,∂f2
∂z (1,2,3) = 6
∂f3
∂x (1,2,3) = 2,∂f3
∂y (1,2,3) = 1,∂f3
∂z (1,2,3) = 6
Therefore, the Jacobian matrix of fat (1,2,3) is:
Jf(1,2,3) =
6−4 2
296
216
Question 4
Question
Let f:R3→R2be a function defined by f(x, y, z)=(x2y, yz). Find the
Jacobian matrix of fat the point (1,2,3).
Solution
To find the Jacobian matrix of fat the point (1,2,3), we need to compute the
partial derivatives of each component function at that point.
Step 1: Compute the partial derivatives. Let f1(x, y, z) = x2yand f2(x, y, z) =
yz. Then, the partial derivatives of fare:
∂f1
∂x = 2xy, ∂f1
∂y =x2,∂f1
∂z = 0
∂f2
∂x = 0,∂f2
∂y =z, ∂f2
∂z =y
Step 2: Evaluate the partial derivatives at the point (1,2,3). At (1,2,3),
we have:
∂f1
∂x (1,2,3) = 4,∂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. The Jacobian matrix of fat (1,2,3)
is given by:
Jf(1,2,3) = "∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #(1,2,3)
=410
032
3
Question 5
Question
Let f:R3→R2be a function defined by f(x, y, z)=(x2y, yz). Determine 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 variable.
Step 1: Calculate the partial derivatives of fLet’s find the partial
derivatives of f:
∂f1
∂x ,∂f1
∂y ,∂f1
∂z ,∂f2
∂x ,∂f2
∂y ,∂f2
∂z
Step 2: Calculate the partial derivatives
∂f1
∂x =∂
∂x (x2y)=2xy
∂f1
∂y =∂
∂y (x2y) = x2
∂f1
∂z = 0
∂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 !=2xy x20
0z y
Therefore, the Jacobian matrix of fis
Jf=2xy x20
0z y
Question 6
Question
Let f:R3→R2be the function defined by f(x, y, z)=(x2−y, 3z). Find 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: Find ∂f1
∂x ,∂f1
∂y , and ∂f1
∂z .
∂f1
∂x =∂
∂x (x2−y)=2x
∂f1
∂y =∂
∂y (x2−y) = −1
∂f1
∂z = 0
Step 2: Find ∂f2
∂x ,∂f2
∂y , and ∂f2
∂z .
∂f2
∂x =∂
∂x (3z) = 0
∂f2
∂y =∂
∂y (3z) = 0
∂f2
∂z =∂
∂z (3z)=3
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 !=2x−1 0
0 0 3
Question 7
Question
Let f:R2→R2be a function defined by f(x, y) = (x2+y2, xy). Determine
the Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we compute the partial derivatives of each
component function of fwith respect to xand y.
Step 1: Compute ∂f1
∂x and ∂f1
∂y for the first component function f1(x, y) =
x2+y2.
∂f1
∂x = 2xand ∂f1
∂y = 2y
Step 2: Compute ∂f2
∂x and ∂f2
∂y for the second component function f2(x, y) =
xy.
∂f2
∂x =yand ∂f2
∂y =x
5
Step 3: Assemble the partial derivatives into a 2 ×2 matrix to obtain the
Jacobian matrix of f.
Jacobian matrix of f="∂f1
∂x
∂f1
∂y
∂f2
∂x
∂f2
∂y #=2x2y
y x
Question 8
Question
Let f:R3→R2be a vector function defined by f(x, y, z)=(x2+y, xy +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.
Step 1: Compute ∂f1
∂x ,∂f1
∂y , and ∂f1
∂z .
∂f1
∂x = 2x
∂f1
∂y = 1
∂f1
∂z = 0
Step 2: Compute ∂f2
∂x ,∂f2
∂y , and ∂f2
∂z .
∂f2
∂x =y
∂f2
∂y =x
∂f2
∂z = 1
Step 3: Assemble the partial derivatives into the Jacobian matrix. The
Jacobian matrix of fis given by
Jf="∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #=2x1 0
y x 1
Question 9
Question
Let f:R3→R2be a function defined by f(x, y, z) = (x2y, yz). Compute the
Jacobian matrix of f.
6
Solution
To find the Jacobian matrix of f, we need to compute the partial derivatives of
fwith respect to each variable and arrange them in a matrix.
Step 1: Compute ∂f1
∂x ,∂f1
∂y , and ∂f1
∂z for the first component of f.
Given f(x, y, z) = (x2y, yz), we have:
f1(x, y, z) = x2y
Calculating the partial derivatives:
∂f1
∂x = 2xy, ∂f1
∂y =x2,∂f1
∂z = 0
Step 2: Compute ∂f2
∂x ,∂f2
∂y , and ∂f2
∂z for the second component of f.
Given f(x, y, z) = (x2y, yz), we have:
f2(x, y, z) = yz
Calculating the partial derivatives:
∂f2
∂x = 0,∂f2
∂y =z, ∂f2
∂z =y
Step 3: Form the Jacobian matrix Jfusing the computed partial deriva-
tives.
The Jacobian matrix Jfof fis given by:
Jf="∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #
Substitute the partial derivatives into the matrix:
Jf=2xy x20
0z y
Therefore, the Jacobian matrix of fis:
Jf=2xy x20
0z y
Question 10
Question
Let f:R3→R3be a transformation defined by f(x, y, z)=(x2+y, y2+z, z2+
x). Find the Jacobian matrix of f.
7
Solution
Step 1: Compute the partial derivatives of fwith respect to x, y, z.
∂f1
∂x = 2x, ∂f1
∂y = 1,∂f1
∂z = 0,
∂f2
∂x = 0,∂f2
∂y = 2y, ∂f2
∂z = 1,
∂f3
∂x = 1,∂f3
∂y = 0,∂f3
∂z = 2z.
Step 2: Assemble the partial derivatives into a 3 ×3 Jacobian matrix.
Jf(x, y, z) =
∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z
∂f3
∂x
∂f3
∂y
∂f3
∂z
=
2x1 0
0 2y1
1 0 2z
.
Therefore, the Jacobian matrix of fis
2x1 0
0 2y1
1 0 2z
.
Question 11
Question
Let f(x, y, z)=(x2+yz, y2+xz, z2+xy) be a vector-valued function. Find the
Jacobian matrix of f.
Solution
Step 1: Compute the partial derivatives of fwith respect to x,y, and z.
∂f
∂x =
∂
∂x (x2+yz)
∂
∂x (y2+xz)
∂
∂x (z2+xy)
=
2x
z
y
Step 2: Compute the partial derivatives of fwith respect to y.
∂f
∂y =
∂
∂y (x2+yz)
∂
∂y (y2+xz)
∂
∂y (z2+xy)
=
z
2y
x
Step 3: Compute the partial derivatives of fwith respect to z.
∂f
∂z =
∂
∂z (x2+yz)
∂
∂z (y2+xz)
∂
∂z (z2+xy)
=
y
x
2z
8
Step 4: Assemble the partial derivatives into the Jacobian matrix of f.
Jf=
2x z y
z2y x
y x 2z
Question 12
Question
Let f:R2→R2be defined by f(x, y) = (3x2+ 2y, x3+y2). Calculate the
Jacobian matrix of fat the point (1,1).
Solution
Step 1: Compute the partial derivatives of fwith respect to xand y.
∂f1
∂x = 6x, ∂f1
∂y = 2
∂f2
∂x = 3x2,∂f2
∂y = 2y
Step 2: Evaluate the partial derivatives at the point (1,1).
∂f1
∂x (1,1) = 6(1) = 6,∂f1
∂y (1,1) = 2
∂f2
∂x (1,1) = 3(1)2= 3,∂f2
∂y (1,1) = 2(1) = 2
Step 3: Assemble the Jacobian matrix by arranging the partial derivatives.
Jf(1,1) = "∂f1
∂x (1,1) ∂f1
∂y (1,1)
∂f2
∂x (1,1) ∂f2
∂y (1,1)#=6 2
3 2
Therefore, the Jacobian matrix of fat the point (1,1) is 6 2
3 2.
Question 13
Question
Let f:R3→R2be a mapping defined by f(x, y, z) = (x2+yz, x +y+z).
Calculate the Jacobian matrix of f.
9
Solution
Step 1: Find the partial derivatives of fwith respect to each variable. The
Jacobian matrix of fis given by:
J="∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #
where f1(x, y, z) = x2+yz and f2(x, y, z) = x+y+z.
Step 2: Find the partial derivatives of f1:
∂f1
∂x = 2x, ∂f1
∂y =z, ∂f1
∂z =y
Step 3: Find the partial derivatives of f2:
∂f2
∂x = 1,∂f2
∂y = 1,∂f2
∂z = 1
Step 4: Assemble the Jacobian matrix Jusing the partial derivatives cal-
culated above:
J=2x z y
1 1 1
Question 14
Question
Let f:R3→R2be a function defined by f(x, y, z)=(x2+yz, y2+xz). Find
the Jacobian matrix of f.
Solution
Step 1: To find the Jacobian matrix of f, we first need to compute the partial
derivatives of each component function.
Step 2: The partial derivatives of fare:
∂f1
∂x = 2x, ∂f1
∂y =z, ∂f1
∂z =y
∂f2
∂x =z, ∂f2
∂y = 2y, ∂f2
∂z =x
Step 3: Using the partial derivatives, we can construct the Jacobian matrix
of f:
Jf="∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #
Jf=2x z y
z2y x
Therefore, the Jacobian matrix of fis 2x z y
z2y x.
10
Question 15
Question
Let f(x, y, z) = x2+ 2y2+ 3z2+ 4xy + 5xz + 6yz. Find the Jacobian matrix of
fat the point (1,1,1).
Solution
Step 1: Compute the partial derivatives of fwith respect to x,y, and z.
∂f
∂x = 2x+ 4y+ 5z
∂f
∂y = 4y+ 6z
∂f
∂z = 6z+ 5x
Step 2: Evaluate the partial derivatives at the point (1,1,1).
∂f
∂x (1,1,1) = 2(1) + 4(1) + 5(1) = 2 + 4 + 5 = 11
∂f
∂y (1,1,1) = 4(1) + 6(1) = 4 + 6 = 10
∂f
∂z (1,1,1) = 6(1) + 5(1) = 6 + 5 = 11
Step 3: Construct the Jacobian matrix of fat (1,1,1). The Jacobian matrix
is a 1 ×3 matrix where the entries are the partial derivatives evaluated at the
point.
Jacobian matrix of fat (1,1,1) : 11 10 11
Question 16
Question
Let f:R3→R2be a function defined by f(x, y, z)=(xy, yz). Determine the
Jacobian matrix of f.
Solution
Step 1: The Jacobian matrix of a function f:Rn→Rmis defined as the matrix
of all first-order partial derivatives of f. The Jacobian matrix of fwill be a
m×nmatrix.
Step 2: Given f(x, y, z)=(xy, yz), let’s find the partial derivatives of f:
∂f1
∂x =y, ∂f1
∂y =x, ∂f1
∂z = 0
11
∂f2
∂x = 0,∂f2
∂y =z, ∂f2
∂z =y
Step 3: The Jacobian matrix of fis:
J="∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #=y x 0
0z y
Question 17
Question
Let f(x, y, z) = xy2z3be a function in R3. Find the Jacobian matrix of fat
the point (1,2,3).
Solution
Step 1: Compute the partial derivatives of fwith respect to each variable.
∂f
∂x =y2z3
∂f
∂y = 2xyz3
∂f
∂z = 3xy2z2
Step 2: Evaluate the partial derivatives at the point (1,2,3).
∂f
∂x = 22·33= 36
∂f
∂y = 2 ·1·33= 54
∂f
∂z = 3 ·22·32= 108
Step 3: Assemble the Jacobian matrix using the partial derivatives. The
Jacobian matrix of fat (1,2,3) is given by:
Jf(1,2,3) = h∂f
∂x
∂f
∂y
∂f
∂z i=36 54 108
Question 18
Question
Let f:R3→R2be the function defined by f(x, y, z) = (x2y, yz). Compute the
Jacobian matrix of f.
12
Solution
Step 1: Compute the partial derivatives of fwith respect to x,y, and z.
∂f
∂x =2xy
0,∂f
∂y =x2
z,∂f
∂z =0
y
Step 2: Assemble the partial derivatives into the Jacobian matrix of f.
Jf= ∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z !=2xy x20
0z y
Question 19
Question
Let f:R3→R2be a function defined by f(x, y, z)=(xy+z, x2−yz). Compute
the Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to compute the partial derivatives of
fwith respect to each input variable x,y, and z.
Step 1: Compute ∂f
∂x
Taking the partial derivative of fwith respect to x, we get:
∂f
∂x =∂
∂x (xy +z)
∂
∂x (x2−yz)=y
2x
Step 2: Compute ∂f
∂y
Taking the partial derivative of fwith respect to y, we get:
∂f
∂y = ∂
∂y (xy +z)
∂
∂y (x2−yz)!=x
−x
Step 3: Compute ∂f
∂z
Taking the partial derivative of fwith respect to z, we get:
∂f
∂z =∂
∂z (xy +z)
∂
∂z (x2−yz)=1
−y
Step 4: Construct the Jacobian matrix J(f)
The Jacobian matrix of fis given by:
J(f) = ∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z !=y x 1
2x−x−y
Therefore, the Jacobian matrix of fis y x 1
2x−x−y.
13
Question 20
Question
Let f:R3→R3be a vector field defined by f(x, y, z) = (xy2, xz, x2y+z).
Determine the Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to compute the matrix of partial
derivatives of fwith respect to x,y, and z.
Step 1: Compute ∂f1∂x, ∂f1
∂y ,∂f1
∂z
∂f1
∂x =∂
∂x (xy2) = y2
∂f1
∂y =∂
∂y (xy2)=2xy
∂f1
∂z =∂
∂z (xy2)=0
Step 2: Compute ∂f2∂x, ∂f2
∂y ,∂f2
∂z
∂f2
∂x =∂
∂x (xz) = z
∂f2
∂y =∂
∂y (xz)=0
∂f2
∂z =∂
∂z (xz) = x
Step 3: Compute ∂f3∂x, ∂f3
∂y ,∂f3
∂z
∂f3
∂x =∂
∂x (x2y+z)=2xy
∂f3
∂y =∂
∂y (x2y+z) = x2
∂f3
∂z =∂
∂z (x2y+z)=1
Step 4: Assemble the Jacobian matrix The Jacobian matrix of fis
Jf=
y22xy 0
z0x
2xy x21
Question 21
Question
Let f:R3→R2be a function defined by f(x, y, z) = (x2+y, y2z). Compute
the Jacobian matrix of fat the point (1,2,3).
14
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 = 2yz, ∂f2
∂z =y2
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) = 2(3) = 6,∂f2
∂z (1,2,3) = 22= 4
Step 3: Assemble the Jacobian matrix of fat (1,2,3).
Jf(1,2,3) = ∂f1
∂x (1,2,3) ∂f1
∂y (1,2,3) ∂f1
∂z (1,2,3)
∂f2
∂x (1,2,3) ∂f2
∂y (1,2,3) ∂f2
∂z (1,2,3)!=210
064
Question 22
Question
Let f:R2→R2be a function defined by f(x, y) = (x2+y, y2−x). Find the
Jacobian matrix of fat the point (1,2).
Solution
Step 1: Compute the partial derivatives of f:
∂f1
∂x = 2x, ∂f1
∂y = 1,
∂f2
∂x =−1,∂f2
∂y = 2y.
Step 2: Evaluate the partial derivatives at the point (1,2):
∂f1
∂x (1,2) = 2,∂f1
∂y (1,2) = 1,
∂f2
∂x (1,2) = −1,∂f2
∂y (1,2) = 4.
Step 3: Construct the Jacobian matrix Jf(1,2):
Jf(1,2) = "∂f1
∂x (1,2) ∂f1
∂y (1,2)
∂f2
∂x (1,2) ∂f2
∂y (1,2)#=2 1
−1 4.
Therefore, the Jacobian matrix of fat (1,2) is 2 1
−1 4.
15
Question 23
Question
Let f(x, y) = x2y
2xy2. Compute the Jacobian matrix Jf(x, y) of fat the point
(1,2).
Solution
Step 1: Compute the partial derivatives of fwith respect to xand y.
∂f
∂x =2xy
2y2,∂f
∂y =x2
4xy
Step 2: Evaluate the partial derivatives at the point (1,2).
∂f
∂x (1,2) = 4
4,∂f
∂y (1,2) = 1
8
Step 3: Construct the Jacobian matrix Jf(1,2).
Jf(1,2) = ∂f1
∂x
∂f1
∂y
∂f2
∂x
∂f2
∂y !=4 1
4 8
Therefore, the Jacobian matrix of fat the point (1,2) is 4 1
4 8.
Question 24
Question
Let f(x, y, z) = xyz2and g(x, y, z) = x2y+y2z+z2x. Find the Jacobian matrix
of the function
h(x, y, z)=(f, g) at the point (1,2,−1).
Solution
Step 1: Compute the partial derivatives of fand gwith respect to x,y, and z.
For f(x, y, z) = xyz2:
∂f
∂x =yz2
∂f
∂y =xz2
∂f
∂z = 2xyz
16
For g(x, y, z) = x2y+y2z+z2x:
∂g
∂x = 2xy +z2
∂g
∂y =x2+ 2yz
∂g
∂z =y2+ 2zx
Step 2: Evaluate the partial derivatives at the point (1,2,−1).
For f:
∂f
∂x = 2
∂f
∂y =−2
∂f
∂z =−4
For g:
∂g
∂x = 4
∂g
∂y = 5
∂g
∂z = 1
Step 3: Assemble the Jacobian matrix of
hat (1,2,−1).
J
h(1,2,−1) = "∂f
∂x
∂f
∂y
∂f
∂z
∂g
∂x
∂g
∂y
∂g
∂z #=2−2−4
4 5 1
Question 25
Question
Let f(x, y, z) = xy2zand g(x, y, z) = x2+y2+z2. Determine the Jacobian
matrix of the composite function h(x, y, z) = f(g(x, y, z), x, y).
Solution
Step 1: Compute the partial derivatives of fwith respect to x,y, and z.
∂f
∂x =y2z, ∂f
∂y = 2xyz, ∂f
∂z =xy2
17
Step 2: Compute the partial derivatives of gwith respect to x,y, and z.
∂g
∂x = 2x, ∂g
∂y = 2y, ∂g
∂z = 2z
Step 3: Express hin terms of fand g.
h(x, y, z) = f(g(x, y, z), x, y) = f(g(x, y, z), x, y) = f(g, x, y) = gxy2
Step 4: Compute the partial derivatives of hwith respect to x,y, and z.
∂h
∂x =y2,∂h
∂y = 2xy, ∂h
∂z = 0
Step 5: Write the Jacobian matrix of h.
Jh=h∂h
∂x
∂h
∂y
∂h
∂z i=y22xy 0
Question 26
Question
Consider the following functions f1(x, y) = x2+ 2yand f2(x, y) = y3−x.
Calculate the Jacobian matrix of the mapping (f1, f2) : R2→R2at the point
(1,2).
Solution
Given functions f1(x, y) = x2+ 2yand f2(x, y) = y3−x, we are asked to
calculate the Jacobian matrix of the mapping (f1, f2) : R2→R2at the point
(1,2).
The Jacobian matrix of a mapping f:Rn→Rmis given by:
J(f) =
∂f1
∂x1
∂f1
∂x2
· · · ∂f1
∂xn
∂f2
∂x1
∂f2
∂x2
· · · ∂f2
∂xn
.
.
..
.
.....
.
.
∂fm
∂x1
∂fm
∂x2
· · · ∂fm
∂xn
In our case, n= 2 and m= 2. So we need to compute the partial derivatives
of f1and f2with respect to xand y.
Step 1: Compute partial derivatives of f1
∂f1
∂x = 2xand ∂f1
∂y = 2
Step 2: Compute partial derivatives of f2
∂f2
∂x =−1 and ∂f2
∂y = 3y2
18
Step 3: Evaluate the Jacobian matrix at (1,2) Substitute x= 1 and
y= 2 into the partial derivatives:
J(f) = 2 2
−1 12
So, the Jacobian matrix of the mapping (f1, f2) at the point (1,2) is:
J(f) = 2 2
−1 12
Question 27
Question
Let f:R3→R2be a mapping defined by f(x, y, z)=(x2+yz, y2+xz). Find
the Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we first need to compute the partial derivatives
of each component function.
Step 1: Find the partial derivatives of f: Let f(x, y, z)=(u, v).
∂u
∂x =∂
∂x (x2+yz)=2x
∂u
∂y =∂
∂y (x2+yz) = z
∂u
∂z =∂
∂z (x2+yz) = y
∂v
∂x =∂
∂x (y2+xz) = z
∂v
∂y =∂
∂y (y2+xz)=2y
∂v
∂z =∂
∂z (y2+xz) = x
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 into the matrix:
19
Jf=2x z y
z2y x
Therefore, the Jacobian matrix of fis
Jf=2x z y
z2y x
.
Question 28
Question
Let f(x) =
x2y2
2xy
z2
. Find the Jacobian matrix Jf(x) of f.
Solution
Step 1: The Jacobian matrix Jf(x) of f(x) is given by:
Jf(x) =
∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z
∂f3
∂x
∂f3
∂y
∂f3
∂z
Step 2: Compute the partial derivatives of f(x): Let f1=x2y2,f2= 2xy,
and f3=z2. We have:
∂f1
∂x = 2xy2,∂f1
∂y = 2x2y, ∂f1
∂z = 0
∂f2
∂x = 2y, ∂f2
∂y = 2x, ∂f2
∂z = 0
∂f3
∂x = 0,∂f3
∂y = 0,∂f3
∂z = 2z
Step 3: Assemble the partial derivatives into the Jacobian matrix:
Jf(x) =
2xy22x2y0
2y2x0
0 0 2z
Therefore, the Jacobian matrix of f(x) is:
Jf(x) =
2xy22x2y0
2y2x0
0 0 2z
20
Question 29
Question
Find the Jacobian matrix of the following transformation:
x=u2+v2
y= 2u−v3
Solution
To find the Jacobian matrix of a transformation, we need to find the partial
derivatives of each component function with respect to each input variable.
Step 1: Find ∂x
∂u ,∂x
∂v ,∂y
∂u ,∂y
∂v .
For x=u2+v2:∂x
∂u = 2u
∂x
∂v = 2v
For y= 2u−v3:
∂y
∂u = 2
∂y
∂v =−3v2
Step 2: Assemble the Jacobian matrix.
The Jacobian matrix is given by:
J=∂x
∂u
∂x
∂v
∂y
∂u
∂y
∂v
Plugging in the partial derivative values we found in Step 1:
J=2u2v
2−3v2
Therefore, the Jacobian matrix of the transformation is:
J=2u2v
2−3v2
Question 30
Question
Let f(x, y, z)=(x3+y3+z3, xyz). Compute the Jacobian matrix of f.
21
Solution
To compute the Jacobian matrix of f, we need to find the partial derivatives of
each component of fwith respect to each input variable.
Step 1: Find ∂f1∂x,∂f1
∂y , and ∂f1
∂z .
∂f1
∂x =∂
∂x (x3+y3+z3)=3x2
∂f1
∂y =∂
∂y (x3+y3+z3)=3y2
∂f1
∂z =∂
∂z (x3+y3+z3)=3z2
Step 2: Find ∂f2∂x,∂f2
∂y , and ∂f2
∂z .
∂f2
∂x =∂
∂x (xyz) = yz
∂f2
∂y =∂
∂y (xyz) = xz
∂f2
∂z =∂
∂z (xyz) = xy
Step 3: Construct the Jacobian matrix. The Jacobian matrix of fis
given by:
Jf="∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #
Therefore, the Jacobian matrix of fis:
Jf=3x23y23z2
yz xz xy
22
