Q: Which of the following is the correct formula for the curvature $$\kappa of a $$space curve $$\vec{r}(t)$$?

$$\kappa = \frac{|\vec{r}'(t) \times \vec{r}''(t)|}{|\vec{r}'(t)|^3}$$

Q: Which of the following is a necessary condition for a vector field $$\vec{F} to be $$conservative in a simply connected region?

$$\nabla \times \vec{F} = \vec{0}$$

Q: Which of the following is the correct statement of Stokes’ Theorem?

$$\iint_S (\nabla \times \vec{F}) \cdot d\vec{S} = \oint_C \vec{F} \cdot d\vec{r}$$

Question 1

Which of the following statements about the cross product of two vectors in $$\mathbb{R}^3 is $$TRUE?

Accepted Answer

The cross product of two vectors is a vector perpendicular to both original vectors.

Question 2

A particle moves in space along the curve given by $$\vec{r}(t) = \langle t, t^2$, t^3$ \rangle. $$What is the speed of the particle at time $$t=2$$?

Accepted Answer

$$\sqrt{1 + 4 + 36} = \sqrt{41}$$

Question 3

Which of the following best describes the gradient vector $$
abla f(x, y) of a $$differentiable function $$f(x, y) at a $$point $$(x_0$, y_0$)$$?

Accepted Answer

It is always perpendicular to the level curve of $$f$$ passing through $$(x_0$, y_0$).$$

Question 4

Suppose $$f(x, y) = x^2$ + y^2$. What is the value of the double integral $$\iint_R f(x, y)\, dA$$ over the region R$ bounded by the circle x^2$ + y^2$ = 1$$?

Accepted Answer

$$\pi$$

Question 5

A vector field $$\vec{F}(x, y) = \langle -y, x \rangle is $$given. Which of the following statements is TRUE?

Accepted Answer

The divergence of $$\vec{F} is $$zero everywhere.

Question 6

Which of the following is the correct formula for the curvature $$\kappa of a $$space curve $$\vec{r}(t)$$?

Accepted Answer

$$\kappa = \frac{|\vec{r}'(t) 	imes \vec{r}''(t)|}{|\vec{r}'(t)|^3}$$

Question 7

A company wants to find the maximum profit $$P(x, y)$$ subject to the constraint $$x + 2y = 10. $$Which method should be used to solve this optimization problem?

Accepted Answer

Use the method of Lagrange multipliers.

Question 8

Evaluate the line integral $$\int_C (x^2$ + y^2$)\, ds$$ where $$C is $$the circle $$x^2$$ + $$y^2 = 4$ traversed once counterclockwise.

Accepted Answer

$$8\pi$$

Question 9

Which of the following is a correct statement of Green’s Theorem?

Accepted Answer

$$\oint_C \vec{F} \cdot d\vec{r} = \iint_R (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y})\, dA$$

Question 10

Suppose $$f(x, y, z) = xyz$$ and $$E is $$the box $$0 \leq x \leq 1$$, $$0 \leq y \leq 2$$, $$0 \leq z \leq 3. $$What is $$\iiint_E f(x, y, z)\, dV$$?

Accepted Answer

$$9$$

Question 11

Which of the following is the correct change of variables formula for a double integral under the transformation $$x = u^2$ - v^2$, y = 2uv$?

Accepted Answer

$$\iint_R f(x, y)\, dxdy = \iint_S f(u^2$ - v^2$, 2uv)\, |J|\, dudv$$

Question 12

A fluid flows with velocity field $$\vec{F}(x, y, z) = \langle x, y, z \rangle. $$What is the divergence of $$\vec{F}$$?

Accepted Answer

$$1 + 1 + 1 = 3$$

Question 13

Which of the following is a necessary condition for a vector field $$\vec{F} to be $$conservative in a simply connected region?

Accepted Answer

$$
abla 	imes \vec{F} = \vec{0}$$

Question 14

Evaluate $$\$$int_0$$^1 \$$int_0$$^1 (x + y)\, dxdy.$$

Accepted Answer

$$1$$

Question 15

Which of the following is the correct statement of Stokes’ Theorem?

Accepted Answer

$$\iint_S (
abla 	imes \vec{F}) \cdot d\vec{S} = \oint_C \vec{F} \cdot d\vec{r}$$

Question 16

A solid hemisphere of radius $$R$$ has constant density $$\rho. $$What is the mass of the hemisphere?

Accepted Answer

$$\frac{2}{3}\pi R^3$ \rho$$

Question 17

Suppose $$f(x, y) = e$$^{x^2 + y^2}$$. What is the value of the directional derivative of f$ at (0$$, 0)$$ in the direction of $$\vec{v} = \langle 3, 4 \rangle$$?

Accepted Answer

$$0$$

Question 18

Which of the following is the correct formula for the surface area of the part of the plane $$z = x + y$$ over the region $$0 \leq x \leq 1$$, $$0 \leq y \leq 1$$?

Accepted Answer

$$\iint_R \sqrt{1 + (f_x)^2$ + (f_y)^2$}\, dA$$

Question 19

Which of the following is the correct statement of the Divergence Theorem?

Accepted Answer

$$\iint_S \vec{F} \cdot d\vec{S} = \iiint_E (
abla \cdot \vec{F})\, dV$$

ENGM2101 Applied Vector Calculus: Course Syllabus and Topic Overview

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