MATH 117 - ELEMENTS OF
MATHEMATICS - Jacobian matrix
Question Bank - Set 1
Liberty University
Question 1
Question
Let f:R3→R2be a function defined by f(x, y, z) = (x2+y, xyz). Find the
Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to compute the partial derivatives of
each component function.
Step 1: Find ∂f1∂x,∂f1
∂y , and ∂f1
∂z .
∂f1
∂x = 2x, ∂f1
∂y = 1,∂f1
∂z = 0.
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.The Jacobian matrix of f
is
Jf="∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #=2x1 0
yz xz xy.
Therefore, the Jacobian matrix of the function fis 2x1 0
yz xz xy.
Question 2
Question
Let f(x, y) = x2yand g(x, y) = exy . Find the Jacobian matrix of the composi-
tion function h(x, y) = g(f(x, y), y).
Solution
Step 1: Find f(x, y) and g(x, y).
f(x, y) = x2y
g(x, y) = exy
Step 2: Find the partial derivatives of f(x, y).
∂f
∂x = 2xy
∂f
∂y =x2
Step 3: Find the value of f(x, y).
f(x, y) = x2y
Step 4: Substitute f(x, y) into g(x, y) to get h(x, y).
h(x, y) = g(f(x, y), y) = ex2y·y=exy2
Step 5: Find the partial derivatives of h(x, y).
∂h
∂x =yexy2
∂h
∂y = 2xyexy2
Step 6: Assemble the Jacobian matrix.
Jacobian matrix of h(x, y) = h∂h
∂x
∂h
∂y i=hyexy22xyexy2i
Question 3
Question
Let f:R3→R2be a function defined by f(x, y, z) = (x2+y, xyz). Find the
Jacobian matrix of fat the point (2,1,−1).
2
Solution
Step 1: To find the Jacobian matrix of fat a point, we need to calculate the
partial derivatives of each component function with respect to each variable at
that point.
Step 2: The components of fare f1(x, y, z) = x2+yand f2(x, y, z) = xyz.
Step 3: Calculating the partial derivatives of f1at the point (2,1,−1):
∂f1
∂x = 2xat (2,1,−1)
∂f1
∂y = 1 at (2,1,−1)
∂f1
∂z = 0 at (2,1,−1)
Step 4: Calculating the partial derivatives of f2at the point (2,1,−1):
∂f2
∂x =yz at (2,1,−1)
∂f2
∂y =xz at (2,1,−1)
∂f2
∂z =xy at (2,1,−1)
Step 5: Assemble the partial derivatives into the Jacobian matrix:
Jf(2,1,−1) = "∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #(2,1,−1)
=410
−1−2−2
Therefore, the Jacobian matrix of fat the point (2,1,−1) is 410
−1−2−2.
Question 4
Question
Let f:R3→R3be the function defined by f(x, y, z) = (x2yz, sin(yz), ex+y).
Find the Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to compute the partial derivatives
of each component function with respect to each variable (x, y, z) and arrange
them in a matrix.
Step 1: Find the partial derivatives of fwith respect to x.
∂f1
∂x =∂
∂x (x2yz)=2xyz
3
∂f2
∂x =∂
∂x (sin(yz)) = 0
∂f3
∂x =∂
∂x (ex+y) = ex+y
Step 2: Find the partial derivatives of fwith respect to y.
∂f1
∂y =∂
∂y (x2yz) = x2z
∂f2
∂y =∂
∂y (sin(yz)) = zcos(yz)
∂f3
∂y =∂
∂y (ex+y) = ex+y
Step 3: Find the partial derivatives of fwith respect to z.
∂f1
∂z =∂
∂z (x2yz) = x2y
∂f2
∂z =∂
∂z (sin(yz)) = ycos(yz)
∂f3
∂z =∂
∂z (ex+y) = 0
Step 4: Assemble the partial derivatives into the Jacobian matrix Jf.
Jf=
2xyz x2z x2y
0zcos(yz)ycos(yz)
ex+yex+y0
Question 5
Question
Find the Jacobian matrix of the following transformation:
f(x, y, z) = ex+y, x2y+z, xyz2
Solution
To find the Jacobian matrix of a transformation, we need to calculate the partial
derivatives of each component function with respect to each input variable.
Step 1: Find ∂f1
∂x ,∂f1
∂y , and ∂f1
∂z for the first component function f1(x, y, z) =
ex+y.∂f1
∂x =ex+y,∂f1
∂y =ex+y,∂f1
∂z = 0
4
Step 2: Find ∂f2
∂x ,∂f2
∂y , and ∂f2
∂z for the second component function f2(x, y, z) =
x2y+z.
∂f2
∂x = 2xy, ∂f2
∂y =x2,∂f2
∂z = 1
Step 3: Find ∂f3
∂x ,∂f3
∂y , and ∂f3
∂z for the third component function f3(x, y, z) =
xyz2.
∂f3
∂x =yz2,∂f3
∂y =xz2,∂f3
∂z = 2xyz
Step 4: Assemble the Jacobian matrix:
Jacobian matrix =
∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z
∂f3
∂x
∂f3
∂y
∂f3
∂z
=
ex+yex+y0
2xy x21
yz2xz22xyz
Question 6
Question
Let f:R3→R3be a function defined by f(x, y, z) = (x2+y2, 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 = 2x, ∂f1
∂y = 2y, ∂f1
∂z = 0
∂f2
∂x = 0,∂f2
∂y =z, ∂f2
∂z =y
∂f3
∂x =z, ∂f3
∂y = 0,∂f3
∂z =x
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) = 2(2) = 4,∂f1
∂z (1,2,3) = 0
∂f2
∂x (1,2,3) = 0,∂f2
∂y (1,2,3) = 3,∂f2
∂z (1,2,3) = 2
∂f3
∂x (1,2,3) = 3,∂f3
∂y (1,2,3) = 0,∂f3
∂z (1,2,3) = 1
Step 3: Assemble the Jacobian matrix of fat (1,2,3) using the partial
derivatives.
Jacobian matrix of fat (1,2,3) =
240
032
301
5
Question 7
Question
Let f:R3→R2be a function defined by f(x, y, z)=(x2+yz, y2+xz).
Determine the Jacobian matrix of f.
Solution
Step 1: To find the Jacobian matrix of f, we first need to compute the partial
derivatives of fwith respect to each variable.
Step 2: The partial derivatives of fare:
∂f
∂x =∂f1
∂x
∂f2
∂x =2x
z
Step 3: Continuing, we find:
∂f
∂y = ∂f1
∂y
∂f2
∂y !=z
2y
Step 4: Lastly, we obtain:
∂f
∂z =∂f1
∂z
∂f2
∂z =y
x
Step 5: Combining the partial derivatives above, we form the Jacobian ma-
trix of f:
Jf(x, y, z) =
2x z
z2y
y x
Therefore, the Jacobian matrix of the function fis
2x z
z2y
y x
.
Question 8
Question
Let f:R3→R2be a function defined by f(x, y, z)=(x2−yz, y2−xz).
Determine the Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to calculate the partial derivatives of
the components of fwith respect to x,y, and z, and arrange them into a 2 ×3
matrix.
6
Step 1: Find ∂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: Find ∂f2∂x,∂f2
∂y , and ∂f2
∂z
∂f2
∂x =∂
∂x (y2−xz) = −z
∂f2
∂y =∂
∂y (y2−xz)=2y
∂f2
∂z =∂
∂z (y2−xz) = −x
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 !=2x−z−y
−z2y−x
Question 9
Question
Let f(x, y, z) = x2y+yz2+xyz. Find the Jacobian matrix of fat the point
(1,2,−1).
Solution
To find the Jacobian matrix of fat the point (1,2,−1), we need to calculate
the partial derivatives of fwith respect to x,y, and zand evaluate them at
(1,2,−1).
Step 1: Calculate ∂f ∂x:
∂f
∂x = 2xy +yz
Step 2: Calculate ∂f ∂y:
∂f
∂y =x2+z2+x
Step 3: Calculate ∂f ∂z:
∂f
∂z = 2yz +xy
7
Step 4: Evaluate the partial derivatives at (1,2,−1):
∂f
∂x (1,2,−1) = 2(1)(2) + (2)(−1) = 2
∂f
∂y (1,2,−1) = (1)2+ (−1)2+ 1 = 1
∂f
∂z (1,2,−1) = 2(2)(−1) + (1)(2) = −2
Step 5: Assemble the Jacobian matrix: The Jacobian matrix Jf(1,2,−1)
is given by:
Jf(1,2,−1) = h∂f
∂x (1,2,−1) ∂f
∂y (1,2,−1) ∂f
∂z (1,2,−1)i
=2 1 −2
Question 10
Question
Let f:R3→R2be a function defined by f(x, y, z) = (x2+y, yz). Find 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: Find ∂f ∂x:
∂f
∂x =∂
∂x (x2+y),∂
∂x (yz)= (2x, 0)
Step 2: Find ∂f ∂y:
∂f
∂y =∂
∂y (x2+y),∂
∂y (yz)= (1, z)
Step 3: Find ∂f ∂z:
∂f
∂z =∂
∂z (x2+y),∂
∂z (yz)= (0, y)
Step 4: Compile into Jacobian matrix: The Jacobian matrix of fis
given by
Jf= ∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z !=2x0 0
1z0
Therefore, the Jacobian matrix of fis 2x0 0
1z0.
8
Question 11
Question
Let f:R3→R2be defined by f(x, y, z) = (x2+yz, xz). Find the Jacobian
matrix of f.
Solution
To find the Jacobian matrix of f, we need to compute the partial derivatives of
each component function and arrange them in a matrix.
Step 1: Find the partial derivatives of f. Let F(x, y, z) = (x2+yz, xz).
The partial derivatives of Fare:
∂F1
∂x = 2x, ∂F1
∂y =z, ∂F1
∂z =y,
∂F2
∂x =z, ∂F2
∂y = 0,∂F2
∂z =x.
Step 2: Assemble the partial derivatives into a Jacobian matrix. The Ja-
cobian matrix of fis given by:
Jf="∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #=2x z y
z0x.
Therefore, the Jacobian matrix of fis 2x z y
z0x.
Question 12
Question
Let f:R3→R2be a differentiable function defined by f(x, y, z)=(x2y, yz).
Find the Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to compute the partial derivatives of
each component function.
Step 1: Find ∂f1
∂x ,∂f1
∂y , and ∂f1
∂z for the first component function f1(x, y, z) =
x2y.
∂f1
∂x = 2xy, ∂f1
∂y =x2,∂f1
∂z = 0
Step 2: Find ∂f2
∂x ,∂f2
∂y , and ∂f2
∂z for the second component function f2(x, y, z) =
yz.
∂f2
∂x = 0,∂f2
∂y =z, ∂f2
∂z =y
9
Step 3: Construct the Jacobian matrix Jfusing the partial derivatives
found above.
Jf="∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #=2xy x20
0z y
Therefore, the Jacobian matrix of fis 2xy x20
0z y.
Question 13
Question
Let f:R3→R2be a function defined by f(x, y, z) = (x2+y2+z2, xyz). Find
the Jacobian matrix of f.
Solution
A Jacobian matrix is a matrix of all first-order partial derivatives of a vector-
valued function. In this case, the Jacobian matrix of f:R3→R2will be a 2 ×3
matrix.
Step 1: Compute the partial derivatives of fLet’s find the partial
derivatives of fwith respect to x,y, and z:
∂f
∂x =∂
∂x (x2+y2+z2)
∂
∂x (xyz)=2x
yz
∂f
∂y = ∂
∂y (x2+y2+z2)
∂
∂y (xyz)!=2y
xz
∂f
∂z =∂
∂z (x2+y2+z2)
∂
∂z (xyz)=2z
xy
Step 2: Assemble the Jacobian matrix The Jacobian matrix is formed
by arranging these partial derivatives as rows:
Jf=2x2y2z
yz xz xy
Question 14
Question
Let f:R2→R2be a function defined by f(x, y) = x
x2+y2,y
x2+y2.
Determine the Jacobian matrix of fat the point (1,1).
10
Solution
Step 1: Compute the partial derivatives of fwith respect to xand y.
∂f1
∂x (x, y) = ∂
∂x x
x2+y2=y2−x2
(x2+y2)2,
∂f1
∂y (x, y) = ∂
∂y x
x2+y2=−2xy
(x2+y2)2,
∂f2
∂x (x, y) = ∂
∂x y
x2+y2=−2xy
(x2+y2)2,
∂f2
∂y (x, y) = ∂
∂y y
x2+y2=x2−y2
(x2+y2)2.
Step 2: Evaluate the partial derivatives at (1,1) to find the Jacobian matrix
of f.
∂f1
∂x (1,1),∂f1
∂y (1,1)=1−1
(1 + 1)2,−2
(1 + 1)2= (0,−1
2),
∂f2
∂x (1,1),∂f2
∂y (1,1)=−2
(1 + 1)2,1−1
(1 + 1)2= (−1
2,0).
Therefore, the Jacobian matrix of fat (1,1) is
0−1
2
−1
20
.
Question 15
Question
Let f:R3→R2be defined by f(x, y, z)=(x2+yz, xyz). Find the Jacobian
matrix Jfof f.
Solution
To find the Jacobian matrix Jfof f, we need to compute the partial derivatives
of each component of fwith respect to each input variable.
Step 1: Find ∂f1
∂x ,
∂f1
∂x =∂
∂x (x2+yz)=2x
Step 2: Find ∂f1
∂y ,
∂f1
∂y =∂
∂y (x2+yz) = z
Step 3: Find ∂f1
∂z ,
∂f1
∂z =∂
∂z (x2+yz) = y
11
Step 4: Find ∂f2
∂x ,
∂f2
∂x =∂
∂x (xyz) = yz
Step 5: Find ∂f2
∂y ,
∂f2
∂y =∂
∂y (xyz) = xz
Step 6: Find ∂f2
∂z ,
∂f2
∂z =∂
∂z (xyz) = xy
Step 7: Construct the Jacobian matrix Jfby arranging the partial deriva-
tives,
Jf="∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #=2x z y
yz xz xy
Question 16
Question
Let f:R3→R3be a vector-valued function defined by f(x, y, z) = (x2+
yz, y2+zx, z2+xy). Find the Jacobian matrix of f.
Solution
Step 1: Compute the partial derivatives of fwith respect to x,y, and z.
∂f
∂x =
2x z y
0z y
0 0 y
,∂f
∂y =
0z x
2y x 0
0x z
,∂f
∂z =
0y x
0 0 x
2z y 0
Step 2: Construct the Jacobian matrix of fby arranging the partial deriva-
tives as columns.
Jf=
2x z y
0z y
0 0 y
Therefore, the Jacobian matrix of fis
Jf=
2x z y
0z y
0 0 y
12
Question 17
Question
Determine the Jacobian matrix for the following system of equations:
(x2+y2= 4
xy = 1
Solution
Step 1: Start by defining the functions f1(x, y) = x2+y2−4 and f2(x, y) = xy−1.
Step 2: Compute the partial derivatives of f1and f2with respect to xand
y:
∂f1
∂x = 2x, ∂f1
∂y = 2y
∂f2
∂x =y, ∂f2
∂y =x
Step 3: Assemble the Jacobian matrix by arranging the partial derivatives:
J="∂f1
∂x
∂f1
∂y
∂f2
∂x
∂f2
∂y #=2x2y
y x
Therefore, the Jacobian matrix for the given system of equations is:
J=2x2y
y x
Question 18
Question
Let f(x, y) = (x2+y, 3y2+x) and g(u, v) = (u+v, u −v). Find the Jacobian
matrix of the composition g◦fat the point (1,2).
Solution
Step 1: Compute the Jacobian matrix of fat the point (1,2).
Let f(x, y)=(x2+y, 3y2+x).
The Jacobian matrix of fis given by:
Jf(x, y) = "∂f1
∂x
∂f1
∂y
∂f2
∂x
∂f2
∂y #=2x1
1 6y.
13
Evaluating at (1,2) gives:
Jf(1,2) = 2 1
1 12.
Step 2: Compute the Jacobian matrix of the composition of g◦fat the
point (1,2).
Let h(u, v) = g(f(u, v)).
By the chain rule, the Jacobian matrix of his given by:
Jh(u, v) = Jg(f(u, v)) ·Jf(u, v).
Step 3: Evaluate the Jacobian matrix at the point (1,2). We have f(1,2) =
(12+ 2,3(2)2+ 1) = (3,13). So the Jacobian matrix Jg(f(1,2)) is the Jacobian
matrix of gevaluated at (3,13):
Jg(3,13) = ∂g1
∂u
∂g1
∂v
∂g2
∂u
∂g2
∂v (3,13)
=1 1
1−1.
Step 4: Calculate the Jacobian matrix of the composition g◦fat the point
(1,2). Finally, we calculate the Jacobian matrix of h=g◦fat (1,2):
Jh(1,2) = Jg(f(1,2)) ·Jf(1,2)
=1 1
1−1·2 1
1 12
=3 11
1−13.
Question 19
Question
Let f(x, y, z)=(x2+y2, y2+z2, z2+x2) be a vector-valued function. Find the
Jacobian matrix of f.
Solution
Step 1: Find the partial derivatives of fwith respect to x, y, and z.
∂f
∂x =
∂
∂x (x2+y2)
∂
∂x (y2+z2)
∂
∂x (z2+x2)
=
2x
0
2x
14
Step 2:
∂f
∂y =
∂
∂y (x2+y2)
∂
∂y (y2+z2)
∂
∂y (z2+x2)
=
2y
2y
0
Step 3:
∂f
∂z =
∂
∂z (x2+y2)
∂
∂z (y2+z2)
∂
∂z (z2+x2)
=
0
2z
2z
Step 4: Assemble the Jacobian matrix by putting the partial derivatives
together.
Jf=
2x2y0
0 2y2z
2x0 2z
Question 20
Question
Let f:R3→R2be defined by f(x, y, z) = (x2y, yz). Compute the Jacobian
matrix of f.
Solution
To compute the Jacobian matrix of f, we need to find the partial derivatives of
each component function.
Step 1: Find ∂f1
∂x ,∂f1
∂y , and ∂f1
∂z for the first component function f1(x, y, z) =
x2y.
∂f1
∂x = 2xy
∂f1
∂y =x2
∂f1
∂z = 0
Step 2: Find ∂f2
∂x ,∂f2
∂y , and ∂f2
∂z for the second component function f2(x, y, z) =
yz.
∂f2
∂x = 0
∂f2
∂y =z
∂f2
∂z =y
15
Step 3: Assemble the Jacobian matrix of fusing the computed partial
derivatives:
Jacobian(f) = "∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #=2xy x20
0z y
Question 21
Question
Let f:R2→R2be a mapping defined by f(x, y) = (e2xcos(y), e2xsin(y)).
Calculate the Jacobian matrix of f.
Solution
Step 1: The Jacobian matrix of fis given by
Jf="∂f1
∂x
∂f1
∂y
∂f2
∂x
∂f2
∂y #.
Step 2: Let’s compute the partial derivatives of fwith respect to xand y.
∂f1
∂x =∂
∂x (e2xcos(y)) = 2e2xcos(y),
∂f1
∂y =∂
∂y (e2xcos(y)) = −e2xsin(y),
∂f2
∂x =∂
∂x (e2xsin(y)) = 2e2xsin(y),
∂f2
∂y =∂
∂y (e2xsin(y)) = e2xcos(y).
Step 3: Now, we can construct the Jacobian matrix of f:
Jf=2e2xcos(y)−e2xsin(y)
2e2xsin(y)e2xcos(y).
Therefore, the Jacobian matrix of fis given by
Jf=2e2xcos(y)−e2xsin(y)
2e2xsin(y)e2xcos(y).
Question 22
Question
Let f:R3→R2be a function defined by f(x, y, z) = (x2+yz, xyz). Find the
Jacobian matrix of fat the point (1,2,−1).
16
Solution
Step 1: First, we find the partial derivatives of fwith respect to x,y, and z:
∂f
∂x =∂(x2+yz)
∂x ,∂(xyz)
∂x = [2x, yz]
∂f
∂y =∂(x2+yz)
∂y ,∂(xyz)
∂y = [z, xz]
∂f
∂z =∂(x2+yz)
∂z ,∂(xyz)
∂z = [y, xy]
Step 2: Next, we can evaluate these partial derivatives at the point (1,2,−1):
∂f
∂x (1,2,−1)
= [2(1),(2)(−1)] = [2,−2]
∂f
∂y (1,2,−1)
= [−1,1]
∂f
∂z (1,2,−1)
= [2,−2]
Step 3: Finally, we construct the Jacobian matrix by arranging these partial
derivatives in rows:
Jacobian matrix of fat (1,2,−1) =
2−2
−1 1
2−2
Question 23
Question
Let f:R3→R2be defined by f(x, y, z)=(x2y, y2z). Find the Jacobian matrix
of f.
Solution
Step 1: We first need to find the partial derivatives of f.
∂f1
∂x = 2xy, ∂f1
∂y =x2,∂f1
∂z = 0
∂f2
∂x = 0,∂f2
∂y = 2yz, ∂f2
∂z =y2
Step 2: Next, we can construct the Jacobian matrix of f.
J(f) = "∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #=2xy x20
0 2yz y2
17
Question 24
Question
Let f:R3→R2be a function defined by f(x, y, z)=(x2+yz, y2+zx). Find
the Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to calculate the partial derivatives of
each component function with respect to each variable and then arrange these
partial derivatives in a matrix.
Step 1: Calculate ∂f1
∂x ,∂f1
∂y , and ∂f1
∂z .
∂f1
∂x = 2x
∂f1
∂y =z
∂f1
∂z =y
Step 2: Calculate ∂f2
∂x ,∂f2
∂y , and ∂f2
∂z .
∂f2
∂x =z
∂f2
∂y = 2y
∂f2
∂z =x
Step 3: Assemble the partial derivatives into the Jacobian matrix of f:
Jacobian matrix of f="∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #=2x z y
z2y x
Therefore, the Jacobian matrix of fis 2x z y
z2y x.
Question 25
Question
Let f(x, y, z) =
2x−3y
x2+y2+z2
xyz
. Find the Jacobian matrix of fat the point
(1,2,−1).
18
Solution
Step 1: Compute the partial derivatives of fwith respect to x,y, and z.
∂f
∂x =
2
2x
yz
,∂f
∂y =
−3
2y
xz
,∂f
∂z =
0
0
xy
Step 2: Evaluate the partial derivatives at the point (1,2,−1).
∂f
∂x (1,2,−1) =
2
2
−2
,∂f
∂y (1,2,−1) =
−3
4
−1
,∂f
∂z (1,2,−1) =
0
0
−2
Step 3: Assemble the Jacobian matrix by arranging the partial derivatives
as columns.
Jf(1,2,−1) =
2−3 0
240
−2−1−2
Question 26
Question
Let f:R3→R2be a continuously differentiable function defined by f(x, y, z) =
(x2y, yz). Find the Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to compute the partial derivatives of
each component function with respect to each input variable.
Step 1: Compute the partial derivatives of f:
∂f1
∂x =∂
∂x (x2y)=2xy,
∂f1
∂y =∂
∂y (x2y) = x2,
∂f1
∂z = 0,
∂f2
∂x = 0,
∂f2
∂y =∂
∂y (yz) = z,
∂f2
∂z =∂
∂z (yz) = y.
19
Step 2: Assemble the Jacobian matrix Jf:
Jf="∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #=2xy x20
0z y.
Therefore, the Jacobian matrix of fis given by 2xy x20
0z y
Question 27
Question
Let f:R3→R2be the function defined by f(x, y, z)=(x2+y, yz). 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 components of f.
Step 2: The components of fare f1(x, y, z) = x2+yand f2(x, y, z) = yz.
We will compute the partial derivatives of these components.
Step 3: The partial derivatives of f1are:
∂f1
∂x = 2x, ∂f1
∂y = 1,∂f1
∂z = 0
Step 4: The partial derivatives of f2are:
∂f2
∂x = 0,∂f2
∂y =z, ∂f2
∂z =y
Step 5: The Jacobian matrix of fis formed by arranging the partial deriva-
tives of f1and f2:
Jf ="∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #=2x1 0
0z y
Question 28
Question
Compute the Jacobian matrix of the following system of equations:
(x2+y2= 4
xy = 1
20
Solution
Step 1: To find the Jacobian matrix, first define the system of equations as a
vector function F(x, y) = (f1(x, y), f2(x, y)) where f1(x, y) = x2+y2−4 and
f2(x, y) = xy −1.
Step 2: Find the partial derivatives of f1and f2with respect to xand y:
∂f1
∂x = 2x, ∂f1
∂y = 2y
∂f2
∂x =y, ∂f2
∂y =x
Step 3: Assemble the Jacobian matrix using these partial derivatives:
J(x, y) = 2x2y
y x
Step 4: Evaluate the Jacobian matrix at the point of interest, here at (x, y) =
(1,1):
J(1,1) = 2(1) 2(1)
1 1 =2 2
1 1
Therefore, the Jacobian matrix of the system at (1,1) is 2 2
1 1.
Question 29
Question
Let f(x, y) = x2+y2
xy −1. Find the Jacobian matrix of fat the point (2,3).
Solution
Step 1: Compute the partial derivatives of fwith respect to xand y.
∂f1
∂x = 2x, ∂f1
∂y = 2y;
∂f2
∂x =y, ∂f2
∂y =x.
Step 2: Evaluate the partial derivatives at the point (2,3).
∂f1
∂x (2,3) = 4,∂f1
∂y (2,3) = 6;
∂f2
∂x (2,3) = 3,∂f2
∂y (2,3) = 2.
Step 3: Construct the Jacobian matrix of fat the point (2,3).
J(f)(2,3) = 4 6
3 2.
21
Question 30
Question
Let f(x, y, z)=(x2+yz, y2+xz, z2+xy). Find the Jacobian matrix of fat
the point (1,2,3).
Solution
Step 1: Calculate the partial derivatives of fwith respect to x,y, and z:
∂f
∂x =
∂f1
∂x
∂f2
∂x
∂f3
∂x
=
2x z y
z0x
y1 0
Step 2: Calculate the partial derivatives of fwith respect to y:
∂f
∂y =
∂f1
∂y
∂f2
∂y
∂f3
∂y
=
z x 0
0 2y z
1z x
Step 3: Calculate the partial derivatives of fwith respect to z:
∂f
∂z =
∂f1
∂z
∂f2
∂z
∂f3
∂z
=
y0x
y x 2z
0 1 y
Step 4: Evaluate each matrix at the point (1,2,3):
∂f
∂x =
232
301
210
∂f
∂y =
310
043
131
∂f
∂z =
201
216
012
Therefore, the Jacobian matrix of fat the point (1,2,3) is:
Jf(1,2,3) =
232
301
210
22
Question 2
Question
Let f(x, y) = x2yand g(x, y) = exy . Find the Jacobian matrix of the composi-
tion function h(x, y) = g(f(x, y), y).
Solution
Step 1: Find f(x, y) and g(x, y).
f(x, y) = x2y
g(x, y) = exy
Step 2: Find the partial derivatives of f(x, y).
∂f
∂x = 2xy
∂f
∂y =x2
Step 3: Find the value of f(x, y).
f(x, y) = x2y
Step 4: Substitute f(x, y) into g(x, y) to get h(x, y).
h(x, y) = g(f(x, y), y) = ex2y·y=exy2
Step 5: Find the partial derivatives of h(x, y).
∂h
∂x =yexy2
∂h
∂y = 2xyexy2
Step 6: Assemble the Jacobian matrix.
Jacobian matrix of h(x, y) = h∂h
∂x
∂h
∂y i=hyexy22xyexy2i
Question 3
Question
Let f:R3→R2be a function defined by f(x, y, z) = (x2+y, xyz). Find the
Jacobian matrix of fat the point (2,1,−1).
2
Solution
Step 1: To find the Jacobian matrix of fat a point, we need to calculate the
partial derivatives of each component function with respect to each variable at
that point.
Step 2: The components of fare f1(x, y, z) = x2+yand f2(x, y, z) = xyz.
Step 3: Calculating the partial derivatives of f1at the point (2,1,−1):
∂f1
∂x = 2xat (2,1,−1)
∂f1
∂y = 1 at (2,1,−1)
∂f1
∂z = 0 at (2,1,−1)
Step 4: Calculating the partial derivatives of f2at the point (2,1,−1):
∂f2
∂x =yz at (2,1,−1)
∂f2
∂y =xz at (2,1,−1)
∂f2
∂z =xy at (2,1,−1)
Step 5: Assemble the partial derivatives into the Jacobian matrix:
Jf(2,1,−1) = "∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #(2,1,−1)
=410
−1−2−2
Therefore, the Jacobian matrix of fat the point (2,1,−1) is 410
−1−2−2.
Question 4
Question
Let f:R3→R3be the function defined by f(x, y, z) = (x2yz, sin(yz), ex+y).
Find the Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to compute the partial derivatives
of each component function with respect to each variable (x, y, z) and arrange
them in a matrix.
Step 1: Find the partial derivatives of fwith respect to x.
∂f1
∂x =∂
∂x (x2yz)=2xyz
3
∂f2
∂x =∂
∂x (sin(yz)) = 0
∂f3
∂x =∂
∂x (ex+y) = ex+y
Step 2: Find the partial derivatives of fwith respect to y.
∂f1
∂y =∂
∂y (x2yz) = x2z
∂f2
∂y =∂
∂y (sin(yz)) = zcos(yz)
∂f3
∂y =∂
∂y (ex+y) = ex+y
Step 3: Find the partial derivatives of fwith respect to z.
∂f1
∂z =∂
∂z (x2yz) = x2y
∂f2
∂z =∂
∂z (sin(yz)) = ycos(yz)
∂f3
∂z =∂
∂z (ex+y) = 0
Step 4: Assemble the partial derivatives into the Jacobian matrix Jf.
Jf=
2xyz x2z x2y
0zcos(yz)ycos(yz)
ex+yex+y0
Question 5
Question
Find the Jacobian matrix of the following transformation:
f(x, y, z) = ex+y, x2y+z, xyz2
Solution
To find the Jacobian matrix of a transformation, we need to calculate the partial
derivatives of each component function with respect to each input variable.
Step 1: Find ∂f1
∂x ,∂f1
∂y , and ∂f1
∂z for the first component function f1(x, y, z) =
ex+y.∂f1
∂x =ex+y,∂f1
∂y =ex+y,∂f1
∂z = 0
4
Step 2: Find ∂f2
∂x ,∂f2
∂y , and ∂f2
∂z for the second component function f2(x, y, z) =
x2y+z.
∂f2
∂x = 2xy, ∂f2
∂y =x2,∂f2
∂z = 1
Step 3: Find ∂f3
∂x ,∂f3
∂y , and ∂f3
∂z for the third component function f3(x, y, z) =
xyz2.
∂f3
∂x =yz2,∂f3
∂y =xz2,∂f3
∂z = 2xyz
Step 4: Assemble the Jacobian matrix:
Jacobian matrix =
∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z
∂f3
∂x
∂f3
∂y
∂f3
∂z
=
ex+yex+y0
2xy x21
yz2xz22xyz
Question 6
Question
Let f:R3→R3be a function defined by f(x, y, z) = (x2+y2, 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 = 2x, ∂f1
∂y = 2y, ∂f1
∂z = 0
∂f2
∂x = 0,∂f2
∂y =z, ∂f2
∂z =y
∂f3
∂x =z, ∂f3
∂y = 0,∂f3
∂z =x
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) = 2(2) = 4,∂f1
∂z (1,2,3) = 0
∂f2
∂x (1,2,3) = 0,∂f2
∂y (1,2,3) = 3,∂f2
∂z (1,2,3) = 2
∂f3
∂x (1,2,3) = 3,∂f3
∂y (1,2,3) = 0,∂f3
∂z (1,2,3) = 1
Step 3: Assemble the Jacobian matrix of fat (1,2,3) using the partial
derivatives.
Jacobian matrix of fat (1,2,3) =
240
032
301
5
Question 7
Question
Let f:R3→R2be a function defined by f(x, y, z)=(x2+yz, y2+xz).
Determine the Jacobian matrix of f.
Solution
Step 1: To find the Jacobian matrix of f, we first need to compute the partial
derivatives of fwith respect to each variable.
Step 2: The partial derivatives of fare:
∂f
∂x =∂f1
∂x
∂f2
∂x =2x
z
Step 3: Continuing, we find:
∂f
∂y = ∂f1
∂y
∂f2
∂y !=z
2y
Step 4: Lastly, we obtain:
∂f
∂z =∂f1
∂z
∂f2
∂z =y
x
Step 5: Combining the partial derivatives above, we form the Jacobian ma-
trix of f:
Jf(x, y, z) =
2x z
z2y
y x
Therefore, the Jacobian matrix of the function fis
2x z
z2y
y x
.
Question 8
Question
Let f:R3→R2be a function defined by f(x, y, z)=(x2−yz, y2−xz).
Determine the Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to calculate the partial derivatives of
the components of fwith respect to x,y, and z, and arrange them into a 2 ×3
matrix.
6
Step 1: Find ∂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: Find ∂f2∂x,∂f2
∂y , and ∂f2
∂z
∂f2
∂x =∂
∂x (y2−xz) = −z
∂f2
∂y =∂
∂y (y2−xz)=2y
∂f2
∂z =∂
∂z (y2−xz) = −x
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 !=2x−z−y
−z2y−x
Question 9
Question
Let f(x, y, z) = x2y+yz2+xyz. Find the Jacobian matrix of fat the point
(1,2,−1).
Solution
To find the Jacobian matrix of fat the point (1,2,−1), we need to calculate
the partial derivatives of fwith respect to x,y, and zand evaluate them at
(1,2,−1).
Step 1: Calculate ∂f ∂x:
∂f
∂x = 2xy +yz
Step 2: Calculate ∂f ∂y:
∂f
∂y =x2+z2+x
Step 3: Calculate ∂f ∂z:
∂f
∂z = 2yz +xy
7
Step 4: Evaluate the partial derivatives at (1,2,−1):
∂f
∂x (1,2,−1) = 2(1)(2) + (2)(−1) = 2
∂f
∂y (1,2,−1) = (1)2+ (−1)2+ 1 = 1
∂f
∂z (1,2,−1) = 2(2)(−1) + (1)(2) = −2
Step 5: Assemble the Jacobian matrix: The Jacobian matrix Jf(1,2,−1)
is given by:
Jf(1,2,−1) = h∂f
∂x (1,2,−1) ∂f
∂y (1,2,−1) ∂f
∂z (1,2,−1)i
=2 1 −2
Question 10
Question
Let f:R3→R2be a function defined by f(x, y, z) = (x2+y, yz). Find 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: Find ∂f ∂x:
∂f
∂x =∂
∂x (x2+y),∂
∂x (yz)= (2x, 0)
Step 2: Find ∂f ∂y:
∂f
∂y =∂
∂y (x2+y),∂
∂y (yz)= (1, z)
Step 3: Find ∂f ∂z:
∂f
∂z =∂
∂z (x2+y),∂
∂z (yz)= (0, y)
Step 4: Compile into Jacobian matrix: The Jacobian matrix of fis
given by
Jf= ∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z !=2x0 0
1z0
Therefore, the Jacobian matrix of fis 2x0 0
1z0.
8
Question 11
Question
Let f:R3→R2be defined by f(x, y, z) = (x2+yz, xz). Find the Jacobian
matrix of f.
Solution
To find the Jacobian matrix of f, we need to compute the partial derivatives of
each component function and arrange them in a matrix.
Step 1: Find the partial derivatives of f. Let F(x, y, z) = (x2+yz, xz).
The partial derivatives of Fare:
∂F1
∂x = 2x, ∂F1
∂y =z, ∂F1
∂z =y,
∂F2
∂x =z, ∂F2
∂y = 0,∂F2
∂z =x.
Step 2: Assemble the partial derivatives into a Jacobian matrix. The Ja-
cobian matrix of fis given by:
Jf="∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #=2x z y
z0x.
Therefore, the Jacobian matrix of fis 2x z y
z0x.
Question 12
Question
Let f:R3→R2be a differentiable function defined by f(x, y, z)=(x2y, yz).
Find the Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to compute the partial derivatives of
each component function.
Step 1: Find ∂f1
∂x ,∂f1
∂y , and ∂f1
∂z for the first component function f1(x, y, z) =
x2y.
∂f1
∂x = 2xy, ∂f1
∂y =x2,∂f1
∂z = 0
Step 2: Find ∂f2
∂x ,∂f2
∂y , and ∂f2
∂z for the second component function f2(x, y, z) =
yz.
∂f2
∂x = 0,∂f2
∂y =z, ∂f2
∂z =y
9
Step 3: Construct the Jacobian matrix Jfusing the partial derivatives
found above.
Jf="∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #=2xy x20
0z y
Therefore, the Jacobian matrix of fis 2xy x20
0z y.
Question 13
Question
Let f:R3→R2be a function defined by f(x, y, z) = (x2+y2+z2, xyz). Find
the Jacobian matrix of f.
Solution
A Jacobian matrix is a matrix of all first-order partial derivatives of a vector-
valued function. In this case, the Jacobian matrix of f:R3→R2will be a 2 ×3
matrix.
Step 1: Compute the partial derivatives of fLet’s find the partial
derivatives of fwith respect to x,y, and z:
∂f
∂x =∂
∂x (x2+y2+z2)
∂
∂x (xyz)=2x
yz
∂f
∂y = ∂
∂y (x2+y2+z2)
∂
∂y (xyz)!=2y
xz
∂f
∂z =∂
∂z (x2+y2+z2)
∂
∂z (xyz)=2z
xy
Step 2: Assemble the Jacobian matrix The Jacobian matrix is formed
by arranging these partial derivatives as rows:
Jf=2x2y2z
yz xz xy
Question 14
Question
Let f:R2→R2be a function defined by f(x, y) = x
x2+y2,y
x2+y2.
Determine the Jacobian matrix of fat the point (1,1).
10
Solution
Step 1: Compute the partial derivatives of fwith respect to xand y.
∂f1
∂x (x, y) = ∂
∂x x
x2+y2=y2−x2
(x2+y2)2,
∂f1
∂y (x, y) = ∂
∂y x
x2+y2=−2xy
(x2+y2)2,
∂f2
∂x (x, y) = ∂
∂x y
x2+y2=−2xy
(x2+y2)2,
∂f2
∂y (x, y) = ∂
∂y y
x2+y2=x2−y2
(x2+y2)2.
Step 2: Evaluate the partial derivatives at (1,1) to find the Jacobian matrix
of f.
∂f1
∂x (1,1),∂f1
∂y (1,1)=1−1
(1 + 1)2,−2
(1 + 1)2= (0,−1
2),
∂f2
∂x (1,1),∂f2
∂y (1,1)=−2
(1 + 1)2,1−1
(1 + 1)2= (−1
2,0).
Therefore, the Jacobian matrix of fat (1,1) is
0−1
2
−1
20
.
Question 15
Question
Let f:R3→R2be defined by f(x, y, z)=(x2+yz, xyz). Find the Jacobian
matrix Jfof f.
Solution
To find the Jacobian matrix Jfof f, we need to compute the partial derivatives
of each component of fwith respect to each input variable.
Step 1: Find ∂f1
∂x ,
∂f1
∂x =∂
∂x (x2+yz)=2x
Step 2: Find ∂f1
∂y ,
∂f1
∂y =∂
∂y (x2+yz) = z
Step 3: Find ∂f1
∂z ,
∂f1
∂z =∂
∂z (x2+yz) = y
11
Step 4: Find ∂f2
∂x ,
∂f2
∂x =∂
∂x (xyz) = yz
Step 5: Find ∂f2
∂y ,
∂f2
∂y =∂
∂y (xyz) = xz
Step 6: Find ∂f2
∂z ,
∂f2
∂z =∂
∂z (xyz) = xy
Step 7: Construct the Jacobian matrix Jfby arranging the partial deriva-
tives,
Jf="∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #=2x z y
yz xz xy
Question 16
Question
Let f:R3→R3be a vector-valued function defined by f(x, y, z) = (x2+
yz, y2+zx, z2+xy). Find the Jacobian matrix of f.
Solution
Step 1: Compute the partial derivatives of fwith respect to x,y, and z.
∂f
∂x =
2x z y
0z y
0 0 y
,∂f
∂y =
0z x
2y x 0
0x z
,∂f
∂z =
0y x
0 0 x
2z y 0
Step 2: Construct the Jacobian matrix of fby arranging the partial deriva-
tives as columns.
Jf=
2x z y
0z y
0 0 y
Therefore, the Jacobian matrix of fis
Jf=
2x z y
0z y
0 0 y
12
Question 17
Question
Determine the Jacobian matrix for the following system of equations:
(x2+y2= 4
xy = 1
Solution
Step 1: Start by defining the functions f1(x, y) = x2+y2−4 and f2(x, y) = xy−1.
Step 2: Compute the partial derivatives of f1and f2with respect to xand
y:
∂f1
∂x = 2x, ∂f1
∂y = 2y
∂f2
∂x =y, ∂f2
∂y =x
Step 3: Assemble the Jacobian matrix by arranging the partial derivatives:
J="∂f1
∂x
∂f1
∂y
∂f2
∂x
∂f2
∂y #=2x2y
y x
Therefore, the Jacobian matrix for the given system of equations is:
J=2x2y
y x
Question 18
Question
Let f(x, y) = (x2+y, 3y2+x) and g(u, v) = (u+v, u −v). Find the Jacobian
matrix of the composition g◦fat the point (1,2).
Solution
Step 1: Compute the Jacobian matrix of fat the point (1,2).
Let f(x, y)=(x2+y, 3y2+x).
The Jacobian matrix of fis given by:
Jf(x, y) = "∂f1
∂x
∂f1
∂y
∂f2
∂x
∂f2
∂y #=2x1
1 6y.
13
Evaluating at (1,2) gives:
Jf(1,2) = 2 1
1 12.
Step 2: Compute the Jacobian matrix of the composition of g◦fat the
point (1,2).
Let h(u, v) = g(f(u, v)).
By the chain rule, the Jacobian matrix of his given by:
Jh(u, v) = Jg(f(u, v)) ·Jf(u, v).
Step 3: Evaluate the Jacobian matrix at the point (1,2). We have f(1,2) =
(12+ 2,3(2)2+ 1) = (3,13). So the Jacobian matrix Jg(f(1,2)) is the Jacobian
matrix of gevaluated at (3,13):
Jg(3,13) = ∂g1
∂u
∂g1
∂v
∂g2
∂u
∂g2
∂v (3,13)
=1 1
1−1.
Step 4: Calculate the Jacobian matrix of the composition g◦fat the point
(1,2). Finally, we calculate the Jacobian matrix of h=g◦fat (1,2):
Jh(1,2) = Jg(f(1,2)) ·Jf(1,2)
=1 1
1−1·2 1
1 12
=3 11
1−13.
Question 19
Question
Let f(x, y, z)=(x2+y2, y2+z2, z2+x2) be a vector-valued function. Find the
Jacobian matrix of f.
Solution
Step 1: Find the partial derivatives of fwith respect to x, y, and z.
∂f
∂x =
∂
∂x (x2+y2)
∂
∂x (y2+z2)
∂
∂x (z2+x2)
=
2x
0
2x
14
Step 2:
∂f
∂y =
∂
∂y (x2+y2)
∂
∂y (y2+z2)
∂
∂y (z2+x2)
=
2y
2y
0
Step 3:
∂f
∂z =
∂
∂z (x2+y2)
∂
∂z (y2+z2)
∂
∂z (z2+x2)
=
0
2z
2z
Step 4: Assemble the Jacobian matrix by putting the partial derivatives
together.
Jf=
2x2y0
0 2y2z
2x0 2z
Question 20
Question
Let f:R3→R2be defined by f(x, y, z) = (x2y, yz). Compute the Jacobian
matrix of f.
Solution
To compute the Jacobian matrix of f, we need to find the partial derivatives of
each component function.
Step 1: Find ∂f1
∂x ,∂f1
∂y , and ∂f1
∂z for the first component function f1(x, y, z) =
x2y.
∂f1
∂x = 2xy
∂f1
∂y =x2
∂f1
∂z = 0
Step 2: Find ∂f2
∂x ,∂f2
∂y , and ∂f2
∂z for the second component function f2(x, y, z) =
yz.
∂f2
∂x = 0
∂f2
∂y =z
∂f2
∂z =y
15
Step 3: Assemble the Jacobian matrix of fusing the computed partial
derivatives:
Jacobian(f) = "∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #=2xy x20
0z y
Question 21
Question
Let f:R2→R2be a mapping defined by f(x, y) = (e2xcos(y), e2xsin(y)).
Calculate the Jacobian matrix of f.
Solution
Step 1: The Jacobian matrix of fis given by
Jf="∂f1
∂x
∂f1
∂y
∂f2
∂x
∂f2
∂y #.
Step 2: Let’s compute the partial derivatives of fwith respect to xand y.
∂f1
∂x =∂
∂x (e2xcos(y)) = 2e2xcos(y),
∂f1
∂y =∂
∂y (e2xcos(y)) = −e2xsin(y),
∂f2
∂x =∂
∂x (e2xsin(y)) = 2e2xsin(y),
∂f2
∂y =∂
∂y (e2xsin(y)) = e2xcos(y).
Step 3: Now, we can construct the Jacobian matrix of f:
Jf=2e2xcos(y)−e2xsin(y)
2e2xsin(y)e2xcos(y).
Therefore, the Jacobian matrix of fis given by
Jf=2e2xcos(y)−e2xsin(y)
2e2xsin(y)e2xcos(y).
Question 22
Question
Let f:R3→R2be a function defined by f(x, y, z) = (x2+yz, xyz). Find the
Jacobian matrix of fat the point (1,2,−1).
16
Solution
Step 1: First, we find the partial derivatives of fwith respect to x,y, and z:
∂f
∂x =∂(x2+yz)
∂x ,∂(xyz)
∂x = [2x, yz]
∂f
∂y =∂(x2+yz)
∂y ,∂(xyz)
∂y = [z, xz]
∂f
∂z =∂(x2+yz)
∂z ,∂(xyz)
∂z = [y, xy]
Step 2: Next, we can evaluate these partial derivatives at the point (1,2,−1):
∂f
∂x (1,2,−1)
= [2(1),(2)(−1)] = [2,−2]
∂f
∂y (1,2,−1)
= [−1,1]
∂f
∂z (1,2,−1)
= [2,−2]
Step 3: Finally, we construct the Jacobian matrix by arranging these partial
derivatives in rows:
Jacobian matrix of fat (1,2,−1) =
2−2
−1 1
2−2
Question 23
Question
Let f:R3→R2be defined by f(x, y, z)=(x2y, y2z). Find the Jacobian matrix
of f.
Solution
Step 1: We first need to find the partial derivatives of f.
∂f1
∂x = 2xy, ∂f1
∂y =x2,∂f1
∂z = 0
∂f2
∂x = 0,∂f2
∂y = 2yz, ∂f2
∂z =y2
Step 2: Next, we can construct the Jacobian matrix of f.
J(f) = "∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #=2xy x20
0 2yz y2
17
Question 24
Question
Let f:R3→R2be a function defined by f(x, y, z)=(x2+yz, y2+zx). Find
the Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to calculate the partial derivatives of
each component function with respect to each variable and then arrange these
partial derivatives in a matrix.
Step 1: Calculate ∂f1
∂x ,∂f1
∂y , and ∂f1
∂z .
∂f1
∂x = 2x
∂f1
∂y =z
∂f1
∂z =y
Step 2: Calculate ∂f2
∂x ,∂f2
∂y , and ∂f2
∂z .
∂f2
∂x =z
∂f2
∂y = 2y
∂f2
∂z =x
Step 3: Assemble the partial derivatives into the Jacobian matrix of f:
Jacobian matrix of f="∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #=2x z y
z2y x
Therefore, the Jacobian matrix of fis 2x z y
z2y x.
Question 25
Question
Let f(x, y, z) =
2x−3y
x2+y2+z2
xyz
. Find the Jacobian matrix of fat the point
(1,2,−1).
18
Solution
Step 1: Compute the partial derivatives of fwith respect to x,y, and z.
∂f
∂x =
2
2x
yz
,∂f
∂y =
−3
2y
xz
,∂f
∂z =
0
0
xy
Step 2: Evaluate the partial derivatives at the point (1,2,−1).
∂f
∂x (1,2,−1) =
2
2
−2
,∂f
∂y (1,2,−1) =
−3
4
−1
,∂f
∂z (1,2,−1) =
0
0
−2
Step 3: Assemble the Jacobian matrix by arranging the partial derivatives
as columns.
Jf(1,2,−1) =
2−3 0
240
−2−1−2
Question 26
Question
Let f:R3→R2be a continuously differentiable function defined by f(x, y, z) =
(x2y, yz). Find the Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to compute the partial derivatives of
each component function with respect to each input variable.
Step 1: Compute the partial derivatives of f:
∂f1
∂x =∂
∂x (x2y)=2xy,
∂f1
∂y =∂
∂y (x2y) = x2,
∂f1
∂z = 0,
∂f2
∂x = 0,
∂f2
∂y =∂
∂y (yz) = z,
∂f2
∂z =∂
∂z (yz) = y.
19
Step 2: Assemble the Jacobian matrix Jf:
Jf="∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #=2xy x20
0z y.
Therefore, the Jacobian matrix of fis given by 2xy x20
0z y
Question 27
Question
Let f:R3→R2be the function defined by f(x, y, z)=(x2+y, yz). 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 components of f.
Step 2: The components of fare f1(x, y, z) = x2+yand f2(x, y, z) = yz.
We will compute the partial derivatives of these components.
Step 3: The partial derivatives of f1are:
∂f1
∂x = 2x, ∂f1
∂y = 1,∂f1
∂z = 0
Step 4: The partial derivatives of f2are:
∂f2
∂x = 0,∂f2
∂y =z, ∂f2
∂z =y
Step 5: The Jacobian matrix of fis formed by arranging the partial deriva-
tives of f1and f2:
Jf ="∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #=2x1 0
0z y
Question 28
Question
Compute the Jacobian matrix of the following system of equations:
(x2+y2= 4
xy = 1
20
Solution
Step 1: To find the Jacobian matrix, first define the system of equations as a
vector function F(x, y) = (f1(x, y), f2(x, y)) where f1(x, y) = x2+y2−4 and
f2(x, y) = xy −1.
Step 2: Find the partial derivatives of f1and f2with respect to xand y:
∂f1
∂x = 2x, ∂f1
∂y = 2y
∂f2
∂x =y, ∂f2
∂y =x
Step 3: Assemble the Jacobian matrix using these partial derivatives:
J(x, y) = 2x2y
y x
Step 4: Evaluate the Jacobian matrix at the point of interest, here at (x, y) =
(1,1):
J(1,1) = 2(1) 2(1)
1 1 =2 2
1 1
Therefore, the Jacobian matrix of the system at (1,1) is 2 2
1 1.
Question 29
Question
Let f(x, y) = x2+y2
xy −1. Find the Jacobian matrix of fat the point (2,3).
Solution
Step 1: Compute the partial derivatives of fwith respect to xand y.
∂f1
∂x = 2x, ∂f1
∂y = 2y;
∂f2
∂x =y, ∂f2
∂y =x.
Step 2: Evaluate the partial derivatives at the point (2,3).
∂f1
∂x (2,3) = 4,∂f1
∂y (2,3) = 6;
∂f2
∂x (2,3) = 3,∂f2
∂y (2,3) = 2.
Step 3: Construct the Jacobian matrix of fat the point (2,3).
J(f)(2,3) = 4 6
3 2.
21
Question 30
Question
Let f(x, y, z)=(x2+yz, y2+xz, z2+xy). Find the Jacobian matrix of fat
the point (1,2,3).
Solution
Step 1: Calculate the partial derivatives of fwith respect to x,y, and z:
∂f
∂x =
∂f1
∂x
∂f2
∂x
∂f3
∂x
=
2x z y
z0x
y1 0
Step 2: Calculate the partial derivatives of fwith respect to y:
∂f
∂y =
∂f1
∂y
∂f2
∂y
∂f3
∂y
=
z x 0
0 2y z
1z x
Step 3: Calculate the partial derivatives of fwith respect to z:
∂f
∂z =
∂f1
∂z
∂f2
∂z
∂f3
∂z
=
y0x
y x 2z
0 1 y
Step 4: Evaluate each matrix at the point (1,2,3):
∂f
∂x =
232
301
210
∂f
∂y =
310
043
131
∂f
∂z =
201
216
012
Therefore, the Jacobian matrix of fat the point (1,2,3) is:
Jf(1,2,3) =
232
301
210
22
Question 2
Question
Let f(x, y) = x2yand g(x, y) = exy . Find the Jacobian matrix of the composi-
tion function h(x, y) = g(f(x, y), y).
Solution
Step 1: Find f(x, y) and g(x, y).
f(x, y) = x2y
g(x, y) = exy
Step 2: Find the partial derivatives of f(x, y).
∂f
∂x = 2xy
∂f
∂y =x2
Step 3: Find the value of f(x, y).
f(x, y) = x2y
Step 4: Substitute f(x, y) into g(x, y) to get h(x, y).
h(x, y) = g(f(x, y), y) = ex2y·y=exy2
Step 5: Find the partial derivatives of h(x, y).
∂h
∂x =yexy2
∂h
∂y = 2xyexy2
Step 6: Assemble the Jacobian matrix.
Jacobian matrix of h(x, y) = h∂h
∂x
∂h
∂y i=hyexy22xyexy2i
Question 3
Question
Let f:R3→R2be a function defined by f(x, y, z) = (x2+y, xyz). Find the
Jacobian matrix of fat the point (2,1,−1).
2
Solution
Step 1: To find the Jacobian matrix of fat a point, we need to calculate the
partial derivatives of each component function with respect to each variable at
that point.
Step 2: The components of fare f1(x, y, z) = x2+yand f2(x, y, z) = xyz.
Step 3: Calculating the partial derivatives of f1at the point (2,1,−1):
∂f1
∂x = 2xat (2,1,−1)
∂f1
∂y = 1 at (2,1,−1)
∂f1
∂z = 0 at (2,1,−1)
Step 4: Calculating the partial derivatives of f2at the point (2,1,−1):
∂f2
∂x =yz at (2,1,−1)
∂f2
∂y =xz at (2,1,−1)
∂f2
∂z =xy at (2,1,−1)
Step 5: Assemble the partial derivatives into the Jacobian matrix:
Jf(2,1,−1) = "∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #(2,1,−1)
=410
−1−2−2
Therefore, the Jacobian matrix of fat the point (2,1,−1) is 410
−1−2−2.
Question 4
Question
Let f:R3→R3be the function defined by f(x, y, z) = (x2yz, sin(yz), ex+y).
Find the Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to compute the partial derivatives
of each component function with respect to each variable (x, y, z) and arrange
them in a matrix.
Step 1: Find the partial derivatives of fwith respect to x.
∂f1
∂x =∂
∂x (x2yz)=2xyz
3
∂f2
∂x =∂
∂x (sin(yz)) = 0
∂f3
∂x =∂
∂x (ex+y) = ex+y
Step 2: Find the partial derivatives of fwith respect to y.
∂f1
∂y =∂
∂y (x2yz) = x2z
∂f2
∂y =∂
∂y (sin(yz)) = zcos(yz)
∂f3
∂y =∂
∂y (ex+y) = ex+y
Step 3: Find the partial derivatives of fwith respect to z.
∂f1
∂z =∂
∂z (x2yz) = x2y
∂f2
∂z =∂
∂z (sin(yz)) = ycos(yz)
∂f3
∂z =∂
∂z (ex+y) = 0
Step 4: Assemble the partial derivatives into the Jacobian matrix Jf.
Jf=
2xyz x2z x2y
0zcos(yz)ycos(yz)
ex+yex+y0
Question 5
Question
Find the Jacobian matrix of the following transformation:
f(x, y, z) = ex+y, x2y+z, xyz2
Solution
To find the Jacobian matrix of a transformation, we need to calculate the partial
derivatives of each component function with respect to each input variable.
Step 1: Find ∂f1
∂x ,∂f1
∂y , and ∂f1
∂z for the first component function f1(x, y, z) =
ex+y.∂f1
∂x =ex+y,∂f1
∂y =ex+y,∂f1
∂z = 0
4
Step 2: Find ∂f2
∂x ,∂f2
∂y , and ∂f2
∂z for the second component function f2(x, y, z) =
x2y+z.
∂f2
∂x = 2xy, ∂f2
∂y =x2,∂f2
∂z = 1
Step 3: Find ∂f3
∂x ,∂f3
∂y , and ∂f3
∂z for the third component function f3(x, y, z) =
xyz2.
∂f3
∂x =yz2,∂f3
∂y =xz2,∂f3
∂z = 2xyz
Step 4: Assemble the Jacobian matrix:
Jacobian matrix =
∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z
∂f3
∂x
∂f3
∂y
∂f3
∂z
=
ex+yex+y0
2xy x21
yz2xz22xyz
Question 6
Question
Let f:R3→R3be a function defined by f(x, y, z) = (x2+y2, 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 = 2x, ∂f1
∂y = 2y, ∂f1
∂z = 0
∂f2
∂x = 0,∂f2
∂y =z, ∂f2
∂z =y
∂f3
∂x =z, ∂f3
∂y = 0,∂f3
∂z =x
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) = 2(2) = 4,∂f1
∂z (1,2,3) = 0
∂f2
∂x (1,2,3) = 0,∂f2
∂y (1,2,3) = 3,∂f2
∂z (1,2,3) = 2
∂f3
∂x (1,2,3) = 3,∂f3
∂y (1,2,3) = 0,∂f3
∂z (1,2,3) = 1
Step 3: Assemble the Jacobian matrix of fat (1,2,3) using the partial
derivatives.
Jacobian matrix of fat (1,2,3) =
240
032
301
5
Question 7
Question
Let f:R3→R2be a function defined by f(x, y, z)=(x2+yz, y2+xz).
Determine the Jacobian matrix of f.
Solution
Step 1: To find the Jacobian matrix of f, we first need to compute the partial
derivatives of fwith respect to each variable.
Step 2: The partial derivatives of fare:
∂f
∂x =∂f1
∂x
∂f2
∂x =2x
z
Step 3: Continuing, we find:
∂f
∂y = ∂f1
∂y
∂f2
∂y !=z
2y
Step 4: Lastly, we obtain:
∂f
∂z =∂f1
∂z
∂f2
∂z =y
x
Step 5: Combining the partial derivatives above, we form the Jacobian ma-
trix of f:
Jf(x, y, z) =
2x z
z2y
y x
Therefore, the Jacobian matrix of the function fis
2x z
z2y
y x
.
Question 8
Question
Let f:R3→R2be a function defined by f(x, y, z)=(x2−yz, y2−xz).
Determine the Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to calculate the partial derivatives of
the components of fwith respect to x,y, and z, and arrange them into a 2 ×3
matrix.
6
Step 1: Find ∂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: Find ∂f2∂x,∂f2
∂y , and ∂f2
∂z
∂f2
∂x =∂
∂x (y2−xz) = −z
∂f2
∂y =∂
∂y (y2−xz)=2y
∂f2
∂z =∂
∂z (y2−xz) = −x
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 !=2x−z−y
−z2y−x
Question 9
Question
Let f(x, y, z) = x2y+yz2+xyz. Find the Jacobian matrix of fat the point
(1,2,−1).
Solution
To find the Jacobian matrix of fat the point (1,2,−1), we need to calculate
the partial derivatives of fwith respect to x,y, and zand evaluate them at
(1,2,−1).
Step 1: Calculate ∂f ∂x:
∂f
∂x = 2xy +yz
Step 2: Calculate ∂f ∂y:
∂f
∂y =x2+z2+x
Step 3: Calculate ∂f ∂z:
∂f
∂z = 2yz +xy
7
Step 4: Evaluate the partial derivatives at (1,2,−1):
∂f
∂x (1,2,−1) = 2(1)(2) + (2)(−1) = 2
∂f
∂y (1,2,−1) = (1)2+ (−1)2+ 1 = 1
∂f
∂z (1,2,−1) = 2(2)(−1) + (1)(2) = −2
Step 5: Assemble the Jacobian matrix: The Jacobian matrix Jf(1,2,−1)
is given by:
Jf(1,2,−1) = h∂f
∂x (1,2,−1) ∂f
∂y (1,2,−1) ∂f
∂z (1,2,−1)i
=2 1 −2
Question 10
Question
Let f:R3→R2be a function defined by f(x, y, z) = (x2+y, yz). Find 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: Find ∂f ∂x:
∂f
∂x =∂
∂x (x2+y),∂
∂x (yz)= (2x, 0)
Step 2: Find ∂f ∂y:
∂f
∂y =∂
∂y (x2+y),∂
∂y (yz)= (1, z)
Step 3: Find ∂f ∂z:
∂f
∂z =∂
∂z (x2+y),∂
∂z (yz)= (0, y)
Step 4: Compile into Jacobian matrix: The Jacobian matrix of fis
given by
Jf= ∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z !=2x0 0
1z0
Therefore, the Jacobian matrix of fis 2x0 0
1z0.
8
Question 11
Question
Let f:R3→R2be defined by f(x, y, z) = (x2+yz, xz). Find the Jacobian
matrix of f.
Solution
To find the Jacobian matrix of f, we need to compute the partial derivatives of
each component function and arrange them in a matrix.
Step 1: Find the partial derivatives of f. Let F(x, y, z) = (x2+yz, xz).
The partial derivatives of Fare:
∂F1
∂x = 2x, ∂F1
∂y =z, ∂F1
∂z =y,
∂F2
∂x =z, ∂F2
∂y = 0,∂F2
∂z =x.
Step 2: Assemble the partial derivatives into a Jacobian matrix. The Ja-
cobian matrix of fis given by:
Jf="∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #=2x z y
z0x.
Therefore, the Jacobian matrix of fis 2x z y
z0x.
Question 12
Question
Let f:R3→R2be a differentiable function defined by f(x, y, z)=(x2y, yz).
Find the Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to compute the partial derivatives of
each component function.
Step 1: Find ∂f1
∂x ,∂f1
∂y , and ∂f1
∂z for the first component function f1(x, y, z) =
x2y.
∂f1
∂x = 2xy, ∂f1
∂y =x2,∂f1
∂z = 0
Step 2: Find ∂f2
∂x ,∂f2
∂y , and ∂f2
∂z for the second component function f2(x, y, z) =
yz.
∂f2
∂x = 0,∂f2
∂y =z, ∂f2
∂z =y
9
Step 3: Construct the Jacobian matrix Jfusing the partial derivatives
found above.
Jf="∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #=2xy x20
0z y
Therefore, the Jacobian matrix of fis 2xy x20
0z y.
Question 13
Question
Let f:R3→R2be a function defined by f(x, y, z) = (x2+y2+z2, xyz). Find
the Jacobian matrix of f.
Solution
A Jacobian matrix is a matrix of all first-order partial derivatives of a vector-
valued function. In this case, the Jacobian matrix of f:R3→R2will be a 2 ×3
matrix.
Step 1: Compute the partial derivatives of fLet’s find the partial
derivatives of fwith respect to x,y, and z:
∂f
∂x =∂
∂x (x2+y2+z2)
∂
∂x (xyz)=2x
yz
∂f
∂y = ∂
∂y (x2+y2+z2)
∂
∂y (xyz)!=2y
xz
∂f
∂z =∂
∂z (x2+y2+z2)
∂
∂z (xyz)=2z
xy
Step 2: Assemble the Jacobian matrix The Jacobian matrix is formed
by arranging these partial derivatives as rows:
Jf=2x2y2z
yz xz xy
Question 14
Question
Let f:R2→R2be a function defined by f(x, y) = x
x2+y2,y
x2+y2.
Determine the Jacobian matrix of fat the point (1,1).
10
Solution
Step 1: Compute the partial derivatives of fwith respect to xand y.
∂f1
∂x (x, y) = ∂
∂x x
x2+y2=y2−x2
(x2+y2)2,
∂f1
∂y (x, y) = ∂
∂y x
x2+y2=−2xy
(x2+y2)2,
∂f2
∂x (x, y) = ∂
∂x y
x2+y2=−2xy
(x2+y2)2,
∂f2
∂y (x, y) = ∂
∂y y
x2+y2=x2−y2
(x2+y2)2.
Step 2: Evaluate the partial derivatives at (1,1) to find the Jacobian matrix
of f.
∂f1
∂x (1,1),∂f1
∂y (1,1)=1−1
(1 + 1)2,−2
(1 + 1)2= (0,−1
2),
∂f2
∂x (1,1),∂f2
∂y (1,1)=−2
(1 + 1)2,1−1
(1 + 1)2= (−1
2,0).
Therefore, the Jacobian matrix of fat (1,1) is
0−1
2
−1
20
.
Question 15
Question
Let f:R3→R2be defined by f(x, y, z)=(x2+yz, xyz). Find the Jacobian
matrix Jfof f.
Solution
To find the Jacobian matrix Jfof f, we need to compute the partial derivatives
of each component of fwith respect to each input variable.
Step 1: Find ∂f1
∂x ,
∂f1
∂x =∂
∂x (x2+yz)=2x
Step 2: Find ∂f1
∂y ,
∂f1
∂y =∂
∂y (x2+yz) = z
Step 3: Find ∂f1
∂z ,
∂f1
∂z =∂
∂z (x2+yz) = y
11
Step 4: Find ∂f2
∂x ,
∂f2
∂x =∂
∂x (xyz) = yz
Step 5: Find ∂f2
∂y ,
∂f2
∂y =∂
∂y (xyz) = xz
Step 6: Find ∂f2
∂z ,
∂f2
∂z =∂
∂z (xyz) = xy
Step 7: Construct the Jacobian matrix Jfby arranging the partial deriva-
tives,
Jf="∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #=2x z y
yz xz xy
Question 16
Question
Let f:R3→R3be a vector-valued function defined by f(x, y, z) = (x2+
yz, y2+zx, z2+xy). Find the Jacobian matrix of f.
Solution
Step 1: Compute the partial derivatives of fwith respect to x,y, and z.
∂f
∂x =
2x z y
0z y
0 0 y
,∂f
∂y =
0z x
2y x 0
0x z
,∂f
∂z =
0y x
0 0 x
2z y 0
Step 2: Construct the Jacobian matrix of fby arranging the partial deriva-
tives as columns.
Jf=
2x z y
0z y
0 0 y
Therefore, the Jacobian matrix of fis
Jf=
2x z y
0z y
0 0 y
12
Question 17
Question
Determine the Jacobian matrix for the following system of equations:
(x2+y2= 4
xy = 1
Solution
Step 1: Start by defining the functions f1(x, y) = x2+y2−4 and f2(x, y) = xy−1.
Step 2: Compute the partial derivatives of f1and f2with respect to xand
y:
∂f1
∂x = 2x, ∂f1
∂y = 2y
∂f2
∂x =y, ∂f2
∂y =x
Step 3: Assemble the Jacobian matrix by arranging the partial derivatives:
J="∂f1
∂x
∂f1
∂y
∂f2
∂x
∂f2
∂y #=2x2y
y x
Therefore, the Jacobian matrix for the given system of equations is:
J=2x2y
y x
Question 18
Question
Let f(x, y) = (x2+y, 3y2+x) and g(u, v) = (u+v, u −v). Find the Jacobian
matrix of the composition g◦fat the point (1,2).
Solution
Step 1: Compute the Jacobian matrix of fat the point (1,2).
Let f(x, y)=(x2+y, 3y2+x).
The Jacobian matrix of fis given by:
Jf(x, y) = "∂f1
∂x
∂f1
∂y
∂f2
∂x
∂f2
∂y #=2x1
1 6y.
13
Evaluating at (1,2) gives:
Jf(1,2) = 2 1
1 12.
Step 2: Compute the Jacobian matrix of the composition of g◦fat the
point (1,2).
Let h(u, v) = g(f(u, v)).
By the chain rule, the Jacobian matrix of his given by:
Jh(u, v) = Jg(f(u, v)) ·Jf(u, v).
Step 3: Evaluate the Jacobian matrix at the point (1,2). We have f(1,2) =
(12+ 2,3(2)2+ 1) = (3,13). So the Jacobian matrix Jg(f(1,2)) is the Jacobian
matrix of gevaluated at (3,13):
Jg(3,13) = ∂g1
∂u
∂g1
∂v
∂g2
∂u
∂g2
∂v (3,13)
=1 1
1−1.
Step 4: Calculate the Jacobian matrix of the composition g◦fat the point
(1,2). Finally, we calculate the Jacobian matrix of h=g◦fat (1,2):
Jh(1,2) = Jg(f(1,2)) ·Jf(1,2)
=1 1
1−1·2 1
1 12
=3 11
1−13.
Question 19
Question
Let f(x, y, z)=(x2+y2, y2+z2, z2+x2) be a vector-valued function. Find the
Jacobian matrix of f.
Solution
Step 1: Find the partial derivatives of fwith respect to x, y, and z.
∂f
∂x =
∂
∂x (x2+y2)
∂
∂x (y2+z2)
∂
∂x (z2+x2)
=
2x
0
2x
14
Step 2:
∂f
∂y =
∂
∂y (x2+y2)
∂
∂y (y2+z2)
∂
∂y (z2+x2)
=
2y
2y
0
Step 3:
∂f
∂z =
∂
∂z (x2+y2)
∂
∂z (y2+z2)
∂
∂z (z2+x2)
=
0
2z
2z
Step 4: Assemble the Jacobian matrix by putting the partial derivatives
together.
Jf=
2x2y0
0 2y2z
2x0 2z
Question 20
Question
Let f:R3→R2be defined by f(x, y, z) = (x2y, yz). Compute the Jacobian
matrix of f.
Solution
To compute the Jacobian matrix of f, we need to find the partial derivatives of
each component function.
Step 1: Find ∂f1
∂x ,∂f1
∂y , and ∂f1
∂z for the first component function f1(x, y, z) =
x2y.
∂f1
∂x = 2xy
∂f1
∂y =x2
∂f1
∂z = 0
Step 2: Find ∂f2
∂x ,∂f2
∂y , and ∂f2
∂z for the second component function f2(x, y, z) =
yz.
∂f2
∂x = 0
∂f2
∂y =z
∂f2
∂z =y
15
Step 3: Assemble the Jacobian matrix of fusing the computed partial
derivatives:
Jacobian(f) = "∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #=2xy x20
0z y
Question 21
Question
Let f:R2→R2be a mapping defined by f(x, y) = (e2xcos(y), e2xsin(y)).
Calculate the Jacobian matrix of f.
Solution
Step 1: The Jacobian matrix of fis given by
Jf="∂f1
∂x
∂f1
∂y
∂f2
∂x
∂f2
∂y #.
Step 2: Let’s compute the partial derivatives of fwith respect to xand y.
∂f1
∂x =∂
∂x (e2xcos(y)) = 2e2xcos(y),
∂f1
∂y =∂
∂y (e2xcos(y)) = −e2xsin(y),
∂f2
∂x =∂
∂x (e2xsin(y)) = 2e2xsin(y),
∂f2
∂y =∂
∂y (e2xsin(y)) = e2xcos(y).
Step 3: Now, we can construct the Jacobian matrix of f:
Jf=2e2xcos(y)−e2xsin(y)
2e2xsin(y)e2xcos(y).
Therefore, the Jacobian matrix of fis given by
Jf=2e2xcos(y)−e2xsin(y)
2e2xsin(y)e2xcos(y).
Question 22
Question
Let f:R3→R2be a function defined by f(x, y, z) = (x2+yz, xyz). Find the
Jacobian matrix of fat the point (1,2,−1).
16
Solution
Step 1: First, we find the partial derivatives of fwith respect to x,y, and z:
∂f
∂x =∂(x2+yz)
∂x ,∂(xyz)
∂x = [2x, yz]
∂f
∂y =∂(x2+yz)
∂y ,∂(xyz)
∂y = [z, xz]
∂f
∂z =∂(x2+yz)
∂z ,∂(xyz)
∂z = [y, xy]
Step 2: Next, we can evaluate these partial derivatives at the point (1,2,−1):
∂f
∂x (1,2,−1)
= [2(1),(2)(−1)] = [2,−2]
∂f
∂y (1,2,−1)
= [−1,1]
∂f
∂z (1,2,−1)
= [2,−2]
Step 3: Finally, we construct the Jacobian matrix by arranging these partial
derivatives in rows:
Jacobian matrix of fat (1,2,−1) =
2−2
−1 1
2−2
Question 23
Question
Let f:R3→R2be defined by f(x, y, z)=(x2y, y2z). Find the Jacobian matrix
of f.
Solution
Step 1: We first need to find the partial derivatives of f.
∂f1
∂x = 2xy, ∂f1
∂y =x2,∂f1
∂z = 0
∂f2
∂x = 0,∂f2
∂y = 2yz, ∂f2
∂z =y2
Step 2: Next, we can construct the Jacobian matrix of f.
J(f) = "∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #=2xy x20
0 2yz y2
17
Question 24
Question
Let f:R3→R2be a function defined by f(x, y, z)=(x2+yz, y2+zx). Find
the Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to calculate the partial derivatives of
each component function with respect to each variable and then arrange these
partial derivatives in a matrix.
Step 1: Calculate ∂f1
∂x ,∂f1
∂y , and ∂f1
∂z .
∂f1
∂x = 2x
∂f1
∂y =z
∂f1
∂z =y
Step 2: Calculate ∂f2
∂x ,∂f2
∂y , and ∂f2
∂z .
∂f2
∂x =z
∂f2
∂y = 2y
∂f2
∂z =x
Step 3: Assemble the partial derivatives into the Jacobian matrix of f:
Jacobian matrix of f="∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #=2x z y
z2y x
Therefore, the Jacobian matrix of fis 2x z y
z2y x.
Question 25
Question
Let f(x, y, z) =
2x−3y
x2+y2+z2
xyz
. Find the Jacobian matrix of fat the point
(1,2,−1).
18
Solution
Step 1: Compute the partial derivatives of fwith respect to x,y, and z.
∂f
∂x =
2
2x
yz
,∂f
∂y =
−3
2y
xz
,∂f
∂z =
0
0
xy
Step 2: Evaluate the partial derivatives at the point (1,2,−1).
∂f
∂x (1,2,−1) =
2
2
−2
,∂f
∂y (1,2,−1) =
−3
4
−1
,∂f
∂z (1,2,−1) =
0
0
−2
Step 3: Assemble the Jacobian matrix by arranging the partial derivatives
as columns.
Jf(1,2,−1) =
2−3 0
240
−2−1−2
Question 26
Question
Let f:R3→R2be a continuously differentiable function defined by f(x, y, z) =
(x2y, yz). Find the Jacobian matrix of f.
Solution
To find the Jacobian matrix of f, we need to compute the partial derivatives of
each component function with respect to each input variable.
Step 1: Compute the partial derivatives of f:
∂f1
∂x =∂
∂x (x2y)=2xy,
∂f1
∂y =∂
∂y (x2y) = x2,
∂f1
∂z = 0,
∂f2
∂x = 0,
∂f2
∂y =∂
∂y (yz) = z,
∂f2
∂z =∂
∂z (yz) = y.
19
Step 2: Assemble the Jacobian matrix Jf:
Jf="∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #=2xy x20
0z y.
Therefore, the Jacobian matrix of fis given by 2xy x20
0z y
Question 27
Question
Let f:R3→R2be the function defined by f(x, y, z)=(x2+y, yz). 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 components of f.
Step 2: The components of fare f1(x, y, z) = x2+yand f2(x, y, z) = yz.
We will compute the partial derivatives of these components.
Step 3: The partial derivatives of f1are:
∂f1
∂x = 2x, ∂f1
∂y = 1,∂f1
∂z = 0
Step 4: The partial derivatives of f2are:
∂f2
∂x = 0,∂f2
∂y =z, ∂f2
∂z =y
Step 5: The Jacobian matrix of fis formed by arranging the partial deriva-
tives of f1and f2:
Jf ="∂f1
∂x
∂f1
∂y
∂f1
∂z
∂f2
∂x
∂f2
∂y
∂f2
∂z #=2x1 0
0z y
Question 28
Question
Compute the Jacobian matrix of the following system of equations:
(x2+y2= 4
xy = 1
20
Solution
Step 1: To find the Jacobian matrix, first define the system of equations as a
vector function F(x, y) = (f1(x, y), f2(x, y)) where f1(x, y) = x2+y2−4 and
f2(x, y) = xy −1.
Step 2: Find the partial derivatives of f1and f2with respect to xand y:
∂f1
∂x = 2x, ∂f1
∂y = 2y
∂f2
∂x =y, ∂f2
∂y =x
Step 3: Assemble the Jacobian matrix using these partial derivatives:
J(x, y) = 2x2y
y x
Step 4: Evaluate the Jacobian matrix at the point of interest, here at (x, y) =
(1,1):
J(1,1) = 2(1) 2(1)
1 1 =2 2
1 1
Therefore, the Jacobian matrix of the system at (1,1) is 2 2
1 1.
Question 29
Question
Let f(x, y) = x2+y2
xy −1. Find the Jacobian matrix of fat the point (2,3).
Solution
Step 1: Compute the partial derivatives of fwith respect to xand y.
∂f1
∂x = 2x, ∂f1
∂y = 2y;
∂f2
∂x =y, ∂f2
∂y =x.
Step 2: Evaluate the partial derivatives at the point (2,3).
∂f1
∂x (2,3) = 4,∂f1
∂y (2,3) = 6;
∂f2
∂x (2,3) = 3,∂f2
∂y (2,3) = 2.
Step 3: Construct the Jacobian matrix of fat the point (2,3).
J(f)(2,3) = 4 6
3 2.
21
Question 30
Question
Let f(x, y, z)=(x2+yz, y2+xz, z2+xy). Find the Jacobian matrix of fat
the point (1,2,3).
Solution
Step 1: Calculate the partial derivatives of fwith respect to x,y, and z:
∂f
∂x =
∂f1
∂x
∂f2
∂x
∂f3
∂x
=
2x z y
z0x
y1 0
Step 2: Calculate the partial derivatives of fwith respect to y:
∂f
∂y =
∂f1
∂y
∂f2
∂y
∂f3
∂y
=
z x 0
0 2y z
1z x
Step 3: Calculate the partial derivatives of fwith respect to z:
∂f
∂z =
∂f1
∂z
∂f2
∂z
∂f3
∂z
=
y0x
y x 2z
0 1 y
Step 4: Evaluate each matrix at the point (1,2,3):
∂f
∂x =
232
301
210
∂f
∂y =
310
043
131
∂f
∂z =
201
216
012
Therefore, the Jacobian matrix of fat the point (1,2,3) is:
Jf(1,2,3) =
232
301
210
22
