Master Multivariable Calculus: Enhance Your Mathematical Skills

Question 1

Which theorem is not a cornerstone of vector calculus?

Accepted Answer

Cauchy-Riemann Equations

Answer

Green's Theorem

Answer

Gauss's Theorem

Answer

Stokes' Theorem

Question 2

State the fundamental principle of the divergence theorem in vector calculus.

Accepted Answer

The divergence theorem links the flux of a vector field across a closed surface to the divergence of the vector field within that surface.

Answer

The divergence theorem is applicable only to vector fields with continuous first-order partial derivatives.

Answer

The divergence theorem facilitates the calculation of the volume enclosed by a closed surface.

Answer

The divergence theorem is a specific instance of Stokes' theorem.

Question 3

Which of the following quantities is NOT a vector?

Accepted Answer

The temperature of a room

Answer

The position of a particle moving in space

Answer

The velocity of a fluid

Answer

The electric field at a point

Question 4

Which of the following is a scalar quantity?

Accepted Answer

The temperature at a point

Answer

The velocity of a fluid

Answer

The position of a particle moving in space

Answer

The electric field at a point

Question 5

Find the gradient of the function f(x, y) = x^2 + y^3 at the point (1, 2).

Accepted Answer

(2, 6)

Answer

(1, 3)

Answer

(6, 2)

Answer

(3, 1)

Question 6

What is the divergence of the vector field F = (P, Q, R)?

Accepted Answer

∇·F = ∂P/∂x + ∂Q/∂y + ∂R/∂z

Answer

∇·F = P∂x + Q∂y + R∂z

Answer

∇·F = P + Q + R

Answer

∇·F = P∂x + Q∂y

Question 7

Find the curl of the vector field F = (x, y, z).

Accepted Answer

(z, -y, x)

Answer

(0, 0, 0)

Answer

(-z, y, -x)

Answer

(x, y, z)

Question 8

Which of the following is a necessary condition for the existence of a potential function for a vector field?

Accepted Answer

The vector field is conservative.

Answer

The curl of the vector field is zero.

Answer

The divergence of the vector field is zero.

Answer

The vector field is continuous.

Question 9

Which of the following formulas calculates the change in a scalar field f(x, y, z) along a curve C parametrized by r(t)?

Accepted Answer

∫[C] (∇f) · r'(t) dt

Answer

∫[C] ∇f · dr

Answer

∫[C] f(x, y, z) ds

Answer

∫[C] (∇f) × r'(t) dt

Question 10

Determine the flux of the vector field F(x, y, z) = (x, y, z) through the surface of the unit sphere centered at the origin.

Accepted Answer

4π

Answer

0

Answer

π

Answer

8π

Question 11

Which of the following is the formula for the gradient of the vector-valued function F(x, y, z) = (f(x, y), g(x, y), h(x, y))?

Accepted Answer

∇F(x, y, z) = (∂f/∂x, ∂g/∂y, ∂h/∂z)

Answer

∇F(x, y, z) = (∂f/∂x, ∂g/∂x, ∂h/∂x)

Answer

∇F(x, y, z) = (∂f/∂x + ∂g/∂y + ∂h/∂z)

Answer

∇F(x, y, z) = (∂f/∂x ⋅ ∂g/∂y ⋅ ∂h/∂z)

Question 12

Find the divergence of the vector field F(x, y, z) = (x^2, y^2, z^2).

Accepted Answer

2

Answer

0

Answer

1

Answer

x^2 + y^2 + z^2

Question 13

Calculate the directional derivative of the function f(x, y) = xy at the point (1, 1) in the direction of the unit vector u = (1/√2, 1/√2).

Accepted Answer

√2

Answer

0

Answer

1

Answer

2

Question 14

Evaluate the double integral ∬R (x^2 + y^2) dA, where R is the region bounded by the curves y = x^2 and y = 2.

Accepted Answer

16/3

Answer

8/3

Answer

32/3

Answer

64/3

Question 15

Which of the following is the direction of maximum rate of change of the function f(x, y, z) = xyz at the point (1, 1, 1)?

Accepted Answer

i + j + k

Answer

-i - j - k

Answer

i - j + k

Answer

-i + j - k

Question 16

What is the value of the surface integral ∬F ⋅ ds over the surface of a sphere of radius R?

Accepted Answer

4πR²

Answer

0

Answer

πR²

Answer

2πR³

Question 17

Which of the following is not a component of a vector-valued function?

Accepted Answer

Scalar

Answer

Magnitude

Answer

Vertical component

Answer

Horizontal component

Question 18

Find the partial derivative of f(x, y) = x^2 + y^3 with respect to x.

Accepted Answer

2x

Answer

2x + 3y^2

Answer

3y^2

Answer

1

Question 19

Which of the following represents the gradient of a scalar field f?

Accepted Answer

∇f

Answer

f∇

Answer

∇·f

Answer

∇xf

Question 20

True or False: The divergence of a conservative vector field is always zero.

Accepted Answer

True

Answer

False

Question 21

Which of the following conditions ensures that a vector field is irrotational?

Accepted Answer

The curl of the vector field is zero.

Answer

The divergence of the vector field is zero.

Answer

The gradient of the vector field is zero.

Answer

The vector field is conservative.

Question 22

Which of the following is true about the divergence theorem?

Accepted Answer

It relates the flux of a vector field through a closed surface to the divergence of the vector field inside the surface.

Answer

It is only applicable to conservative vector fields.

Answer

It is equivalent to Green's Theorem for three-dimensional regions.

Answer

It is not applicable to surfaces with boundaries.

Question 23

Which of the following is equal to the gradient of a scalar field?

Accepted Answer

A vector field

Answer

A scalar field

Answer

A vector-valued function

Answer

A matrix

Question 24

What is the formula for the divergence of a vector field F(x, y, z) = (P, Q, R)?

Accepted Answer

Div(F) = ∂P/∂x + ∂Q/∂y + ∂R/∂z

Answer

Div(F) = (∂P/∂x, ∂Q/∂y, ∂R/∂z)

Answer

Div(F) = P + Q + R

Answer

Div(F) = ||P|| + ||Q|| + ||R||

Question 25

Which of the following is **NOT** a type of multiple integral?

Accepted Answer

Curvilinear integral

Answer

Double integral

Answer

Triple integral

Answer

Volume integral

Question 26

Which of the following integral theorems is used to calculate the flux of a vector field through a surface?

Accepted Answer

Gauss' Theorem

Answer

Green's Theorem

Answer

Stokes' Theorem

Answer

Fundamental Theorem of Calculus

Question 27

What is the physical interpretation of the curl of a vector field?

Accepted Answer

It represents the circulation of the vector field around a point.

Answer

It represents the divergence of the vector field.

Answer

It represents the gradient of the vector field.

Answer

It represents the work done by the vector field.

Question 28

Which of the following is the surface area of the surface given by z = f(x, y) over the region R in the xy-plane?

Accepted Answer

∬R (1 + (∂f/∂x)^2 + (∂f/∂y)^2) dA

Answer

∬R (1 - (∂f/∂x)^2 + (∂f/∂y)^2) dA

Answer

∬R ((∂f/∂x)^2 + (∂f/∂y)^2) dA

Answer

∬R (1 + (∂f/∂x)^2 - (∂f/∂y)^2) dA

Question 29

Which of the following is the curl of the vector field F(x, y, z) = (x, y, z)?

Accepted Answer

(0, 0, 0)

Answer

(x, y, z)

Answer

(y, -x, 0)

Answer

(z, -y, x)

Question 30

Which of the following is an example of a scalar potential for the vector field F(x, y, z) = (2x, 3y, 4z)?

Accepted Answer

x^2 + y^3 + z^4

Answer

2x^2 + 3y^2 + 4z^2

Answer

x + y + z

Answer

xyz

Question 31

Find the divergence of the vector field F = (e^x * cos(y))i + (e^x * sin(y))j in 2D:

Accepted Answer

e^x

Answer

-e^x

Answer

0

Answer

1

Question 32

Find the surface integral ∬[S] F . dS, where F = (x + z)i + (y + x)j + (z + y)k and S is the surface of the sphere x^2 + y^2 + z^2 = 1 with upward orientation:

Accepted Answer

4π

Answer

2π

Answer

8π

Answer

0

Question 33

Use the Chain Rule to compute the partial derivative ∂z/∂x for the function f(x, y) = sin(x^2 + y^2):

Accepted Answer

2x * cos(x^2 + y^2)

Answer

cos(x^2 + y^2)

Answer

sin(x^2 + y^2)

Answer

-2x * cos(x^2 + y^2)

Question 34

Calculate the Jacobian of the transformation x = u + v, y = u - v.

Accepted Answer

2

Answer

1

Answer

-2

Answer

-1

Question 35

Determine the surface area of the part of the surface z = xy that lies inside the cylinder x^2 + y^2 = 1.

Accepted Answer

π/4

Answer

π/2

Answer

π

Answer

2π

Question 36

Which of the following integrals represents the work done by the force field F(x, y) = (x^2, y^3) in moving an object along the curve C given by the parametric equations x = cos(t), y = sin(t) for 0 ≤ t ≤ π?

Accepted Answer

∫C F · dr = ∫0^π (cos^3(t) - sin^3(t)) dt

Answer

∫C F · dr = ∫0^π (cos^2(t) + sin^2(t)) dt

Answer

∫C F · dr = ∫0^π (y^3 - x^2) dt

Answer

∫C F · dr = ∫0^π (x^2 y^3) dt

Question 37

Find the gradient of the vector-valued function F(x, y, z) = (x^2 + y^2)i + (y^2 + z^2)j + (z^2 + x^2)k.

Accepted Answer

2xi + 2yj + 2zk

Answer

x^2i + y^2j + z^2k

Answer

xi + yj + zk

Question 38

Determine the directional derivative of the function f(x, y) = sin(x + y) at the point (π/4, π/4) in the direction of the vector **u** = (1/√2, 1/√2).

Accepted Answer

√2/2

Answer

0

Answer

1

Answer

√2

Question 39

Calculate the triple integral of the function f(x, y, z) = x + y + z over the solid region E. The region E is bounded by the coordinate planes and the plane defined by the equation x + y + z = 3.

Accepted Answer

9

Answer

3

Answer

18

Answer

27

Question 40

Determine the divergence of the vector field F(x, y, z) = (x^2 + y, y^2 + z, z^2 + x).

Accepted Answer

2x + 2y + 2z

Answer

x^2 + y^2 + z^2

Answer

1

Answer

0

Question 41

What is the equation of the tangent plane to the surface defined by the equation z = xy + x^2 at the point (1, 1, 2)?

Accepted Answer

z = 2 + x + y

Answer

z = 2 + 2x + y

Answer

z = 3 + x + y

Answer

z = 1 + x + y

Question 42

Use the method of Lagrange multipliers to find the maximum and minimum values of the function f(x, y) = x^2 + y^2, subject to the constraint x^2 + y^2 = 4.

Accepted Answer

Maximum: (2, 0) and (-2, 0); Minimum: (0, 2) and (0, -2)

Answer

Maximum: (0, 2) and (0, -2); Minimum: (2, 0) and (-2, 0)

Answer

Maximum: (1, 1) and (-1, -1); Minimum: (1, -1) and (-1, 1)

Question 43

Which of the following expressions does NOT represent a vector-valued function?

Accepted Answer

x^2 + y^2

Answer

r(t) = <t, t^2, t^3>

Answer

(cos(t), sin(t))

Answer

r(t) = ti + tj

Question 44

Evaluate the line integral of the vector field F(x, y) = (y, x) along the curve C parameterized by r(t) = (t, t^2) from t = 0 to t = 1.

Accepted Answer

1/2

Answer

0

Answer

1

Answer

-1

Question 45

Calculate the double integral of the function f(x, y) = e^(x+y) over the region R in the xy-plane bounded by the lines y = 0, y = x, and x = 1.

Accepted Answer

e - 1

Answer

-e + 1

Answer

-e - 1

Answer

e + 1

Question 46

What is the flux of the vector field F(x, y, z) = (x, y, z) across the surface of the unit sphere?

Accepted Answer

4π/3

Answer

π

Answer

0

Answer

2π

Question 47

Identify the potential function for the force field F(x, y) = (y, -x).

Accepted Answer

xy

Answer

x^2 + y^2

Answer

e^(xy)

Answer

sin(xy)