Master Calculus III: Vector Calculus and Multivariable Calculus for Mechanical Engineering

Question 1

Calculate the divergence of the vector field (f{F}(x,y,z)) = x(hat{f{i}}) + y^2(hat{f{j}}) + z^3(hat{f{k}}).

Accepted Answer

1+2y+3z

Answer

x+y^2+z^3

Answer

0

Answer

1

Question 2

Which of the following is a rule for differentiating vector functions, rather than a fundamental theorem of vector calculus?

Accepted Answer

The Chain Rule for Vector Functions

Answer

Green's Theorem

Answer

Gauss's Theorem

Answer

Stokes' Theorem

Question 3

Determine the velocity vector of the differentiable vector function r(t) = <x(t), y(t), z(t)>.

Accepted Answer

The vector (dx/dt, dy/dt, dz/dt)

Answer

The vector (dy/dt, dx/dt, dz/dt)

Answer

The vector (x(t), y(t), z(t))

Question 4

Which theorem in vector calculus enables you to convert a surface integral over a closed surface into a volume integral within the enclosed region?

Accepted Answer

Gauss's Theorem (Divergence Theorem)

Answer

Stokes' Theorem

Question 5

Find the value of the line integral ∫[C] (x^2 + y) dx + (y - x) dy, where the curve C is given by y = x^3 from (0, 0) to (1, 1).

Accepted Answer

1/2

Answer

1

Answer

0

Question 6

Which theorem relates the surface integral of a vector field over a closed surface to the volume integral of the divergence of the vector field within the enclosed region?

Accepted Answer

Divergence Theorem

Answer

Stokes' Theorem

Answer

Fundamental Theorem of Line Integrals

Answer

Green's Theorem

Question 7

Evaluate the flux integral ∬_S F · dS using the divergence theorem, where F(x, y, z) = x^2 i + y^2 j + z^2 k, and S is the part of the paraboloid z = 4 - x^2 - y^2 above the xy-plane.

Accepted Answer

16π

Answer

0

Answer

32π

Question 8

Determine if the vector field F = (2xy + z)i + (x² + 3z)j + (x + 2y)k is conservative. If it is, find a potential function f such that ∇f = F.

Accepted Answer

The vector field is conservative, and a potential function is f(x, y, z) = x²y + 3xz + y² + 2xy + C.

Answer

The vector field is not conservative.

Answer

The vector field is conservative, and a potential function is f(x, y, z) = xy + z + C.

Question 9

The line integral of a vector field represents the:

Accepted Answer

Work done by the field over a curve

Answer

Circulation of the field around a closed curve

Answer

Flux of the field through a surface

Answer

Divergence of the field at a point

Question 10

The physical interpretation of the curl of a vector field is:

Accepted Answer

Circulation per unit area

Answer

Flux per unit volume

Answer

Divergence per unit length

Answer

Gradient per unit time

Question 11

Which of the following is NOT a type of vector field?

Accepted Answer

Vector field

Answer

Gradient field

Answer

Conservative field

Answer

Solenoidal field

Question 12

The curl of a vector field F is a vector field that measures the:

Accepted Answer

Circulation of the vector field around a small loop

Answer

Rate of change of the vector field in the direction of its flow

Answer

Amount of outward flux of the vector field through a small surface

Answer

Divergence of the vector field

Question 13

In the context of a vector field F along a curve C, what is the definition of a line integral?

Accepted Answer

The integral of F's dot product with C's tangent vector

Answer

The integral of F's cross product with C's tangent vector

Answer

The integral of F's scalar component in the direction of C

Answer

The integral of F's vector component in the direction of C

Question 14

What is the Hessian matrix of a function?

Accepted Answer

A matrix containing a function's second partial derivatives as its elements

Answer

A matrix containing a function's first partial derivatives as its elements

Answer

A matrix containing a function's mixed partial derivatives as its elements

Answer

A matrix containing a function's directional derivatives as its elements

Question 15

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

Accepted Answer

4/3

Answer

2/3

Answer

8/3

Answer

16/3

Question 16

Find the partial derivatives of the function f(x, y) = e^(x^2 + y^2).

Accepted Answer

fx = 2x e^(x^2 + y^2), fy = 2y e^(x^2 + y^2)

Answer

fx = xe^(x^2 + y^2), fy = ye^(x^2 + y^2)

Answer

fx = e^(x^2 + y^2), fy = 2y e^(x^2 + y^2)

Answer

fx = 2xe^(x^2 + y^2), fy = e^(x^2 + y^2)

Question 17

Determine whether the vector field F(x, y, z) = (y - z, z - x, x - y) is conservative.

Accepted Answer

Conservative

Answer

Not conservative

Question 18

Find the partial derivatives of the function f(x, y, z) = x^2y + yz^3 with respect to x, y, and z.

Accepted Answer

2xy + z^3, x^2 + 3yz^2, y + 3xz^2

Answer

2xy - z^3, x^2 + 3yz^2, y - 3xz^2

Answer

2xy + z^3, x^2 - 3yz^2, y + 3xz^2

Answer

x^2 + z^3, x^2 + 3yz^2, y - 3xz^2

Question 19

Determine if the vector field F(x, y) = (y, -x) is conservative.

Accepted Answer

Yes

Answer

Only in certain regions

Answer

No

Answer

Cannot be determined

Question 20

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

Accepted Answer

e√2

Answer

e

Answer

0

Answer

e/2

Question 21

Use Green's Theorem to evaluate the line integral ∫C (xy + y^2) dx + (x^2 + 2y) dy, where C is the boundary of the quarter-circle in the first quadrant bounded by the curves y = x^2 and y = √x..

Accepted Answer

π/12

Answer

-π/3

Answer

π/3

Answer

-π/12

Question 22

What is the formula for calculating the line integral of a vector field F(x, y, z) along a smooth curve r(t) from point A to point B?

Accepted Answer

∫[A,B] F(r(t)) ⋅ r'(t) dt

Answer

∫[A,B] r'(t) ⋅ F(r(t)) dt

Answer

∫[A,B] r(t) ⋅ F(r(t)) dt

Question 23

Which theorem relates the surface integral of a vector field F over a smooth surface S to the volume integral of its divergence over the enclosed region E?

Accepted Answer

Divergence Theorem

Answer

Gauss's Theorem

Answer

Stokes' Theorem

Answer

Green's Theorem

Question 24

Calculate the volume of the solid bounded by the surfaces z = x^2 + y^2 and z = 4 using multiple integrals.

Accepted Answer

∫∫[x^2+y^2<=4] 4 - x^2 - y^2 dx dy

Answer

∫∫[x^2+y^2>=4] x^2 + y^2 dx dy

Answer

∫∫[x^2+y^2>=4] 4 - x^2 - y^2 dx dy

Answer

∫∫[x^2+y^2<=4] x^2 + y^2 dx dy

Question 25

Find a formula for calculating the surface area of a smooth parametrized surface given by r(u, v) over a region D in the uv-plane.

Accepted Answer

∫∫[D] || r_u x r_v || dA

Answer

∫∫[D] r_u + r_v dA

Answer

∫∫[D] r_u ⋅ r_v dA

Answer

∫∫[D] (r_u x r_v) ⋅ dA

Question 26

Find the formula for calculating the divergence of a vector field F = (P, Q, R).

Accepted Answer

∇ ⋅ F = (∂P/∂x) + (∂Q/∂y) + (∂R/∂z)

Answer

∇ ⋅ F = (∂P/∂y) + (∂Q/∂x) + (∂R/∂z)

Answer

∇ ⋅ F = (∂P/∂z) + (∂Q/∂x) + (∂R/∂y)

Answer

∇ ⋅ F = (∂P/∂y) + (∂Q/∂z) + (∂R/∂x)

Question 27

Determine the formula for finding the gradient of a scalar field f(x, y, z).

Accepted Answer

∇f = (∂f/∂x) i + (∂f/∂y) j + (∂f/∂z) k

Answer

∇f = (∂f/∂y) i + (∂f/∂z) j + (∂f/∂x) k

Answer

∇f = (∂f/∂x) j + (∂f/∂y) k + (∂f/∂z) i

Answer

∇f = (∂f/∂z) i + (∂f/∂x) j + (∂f/∂y) k

Question 28

Which of the following is NOT a theorem related to vector calculus?

Accepted Answer

Pythagorean Theorem

Answer

Gauss's Theorem

Answer

Green's Theorem

Answer

Stokes' Theorem

Question 29

Calculate the line integral of the vector field F(x, y) = (x^2 + y^2, x - y) along the curve C given by the vector function r(t) = (t, t^2) for 0 ≤ t ≤ 1.

Accepted Answer

1/2

Answer

2

Answer

0

Answer

1

Question 30

What is the divergence of the vector field F(x, y, z) = (x^2 - y^2, y^2 - z^2, z^2 - x^2)?

Accepted Answer

0

Answer

2

Answer

1

Answer

-1

Question 31

Which of the following is NOT a method for evaluating a line integral?

Accepted Answer

Method of Residues

Answer

Green's Theorem

Answer

Parametrization

Answer

Stokes' Theorem

Question 32

Which of the following is NOT a concept related to vector calculus?

Accepted Answer

Laplace transform

Answer

Circulation

Answer

Flux

Answer

Gradient

Question 33

Find the line integral of the vector field F(x, y) = (x^2 + y^2, 2xy) along the curve C given by the vector function r(t) = (t, t^2) for 0 ≤ t ≤ 1.

Accepted Answer

∫[0,1] (t^2 + t^4, 4t^3) dt

Answer

∫[0,1] (t^3 + t^5, 4t^4) dt

Answer

∫[0,1] (t + t^2, 4t^3) dt

Answer

∫[0,1] (x^2 + y^2, 2xy) dt

Question 34

Calculate the surface integral ∬[S] (x + y) dS, where S is the part of the plane z = 2x + y that lies inside the cylinder x^2 + y^2 = 1.

Accepted Answer

∫[0,2π] ∫[0,1] (r cos θ + r sin θ) r dr dθ

Answer

∫[0,2π] ∫[0,2] (x + y) dx dy

Answer

∫[0,2π] ∫[0,1] (r cos θ + r sin θ) dr dθ

Answer

∫[0,1] ∫[0,2] (x + y) dx dy

Question 35

Use the Divergence Theorem to compute the flux of the vector field F(x, y, z) = (x, y, z) through the surface of the solid region bounded by the paraboloid z = x^2 + y^2 and the plane z = 4.

Accepted Answer

∫∫[D] (2x + 2y) dA = 16π

Answer

∫∫[D] (x + y + z) dA = 16π

Answer

∫∫[D] (x^2 + y^2) dA = 16π

Answer

∫∫[D] z dA = 16π

Question 36

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

Accepted Answer

2xy + 4y^2

Answer

x^3 + 2xy

Answer

4xy

Answer

3x^2 + 2y

Question 37

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

Accepted Answer

∫[0,1] ∫[x,x^2] (x^2 + y^2) dy dx

Answer

∫[0,1] ∫[x^2,x] (x^2 + y^2) dy dx

Answer

∫[0,1] (x^2 + x^4) dx dy

Answer

∫[0,1] (x^2 + y^2) dx dy

Question 38

Determine the volume of the solid region lying below the surface z = xy and above the triangle in the xy-plane bounded by the lines y = 0, x = 2, and y = x.

Accepted Answer

∫[0,2] ∫[0,x] xy dy dx = 4/3

Answer

∫[0,2] y^2 dx dy = 4/3

Answer

∫[0,2] ∫[x,2] xy dy dx = 4/3

Answer

∫[0,2] x^2 dy dx = 4/3

Question 39

Find the gradient of the function f(x, y, z) = sin(x + y + z).

Accepted Answer

(cos(x + y + z), cos(x + y + z), cos(x + y + z))

Answer

(x + y + z, x + y + z, x + y + z)

Answer

(sin(x + y + z), sin(x + y + z), sin(x + y + z))

Answer

(x, y, z)

Question 40

Consider the vector field F(x, y) = (y, -x). Find a potential function for F.

Accepted Answer

f(x, y) = xy

Answer

f(x, y) = x + y

Answer

f(x, y) = x^2 + y^2

Answer

f(x, y) = y - x

Question 41

Which of the following is the correct formula for the curl of a vector field F(x, y, z)?

Accepted Answer

∇ x F = (∂Fz/∂y - ∂Fy/∂z, ∂Fx/∂z - ∂Fz/∂x, ∂Fy/∂x - ∂Fx/∂y)

Answer

∇ x F = (∂Fz/∂y + ∂Fy/∂z, ∂Fx/∂z + ∂Fz/∂y, ∂Fy/∂x + ∂Fx/∂z)

Answer

∇ x F = (∂Fz/∂x + ∂Fy/∂z, ∂Fx/∂y + ∂Fz/∂x, ∂Fy/∂z + ∂Fx/∂y)