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

$$\iint_S \nabla \times \vec{F} \cdot d\vec{S} = \int_C \vec{F} \cdot d\vec{r}$$

Question 1

Which of the following statements correctly describes the geometric meaning of the dot product $$\vec{a} \cdot \vec{b} of $$two vectors $$\vec{a}$$ and $$\vec{b} in$$ $$\mathbb{R}^3$$?

Accepted Answer

It gives the product of the magnitudes of $$\vec{a}$$ and $$\vec{b}$$ times the cosine of the angle between them.

Question 2

Suppose a plane in $$\mathbb{R}^3$$ passes through the point $$(1,2,3)$$ and has normal vector $$\vec{n} = \langle 2, -1, 4 \rangle. $$Which of the following is the correct equation for the plane?

Accepted Answer

$$2(x-1) - (y-2) + 4(z-3) = 0$$

Question 3

Given the vector-valued function $$\vec{r}(t) = \langle t^2$, \sin t, e^t \rangle$$, what is $$\vec{r}'(t)$$?

Accepted Answer

$$\langle 2t, \cos t, e^t \rangle$$

Question 4

A particle moves along the curve $$\vec{r}(t) = \langle t, t^2$, t^3$ \rangle$$ for $$t \in [0,1]. $$What is the length of the curve traversed by the particle?

Accepted Answer

$$\$$int_0$$^1 \sqrt{1 + 4t^2$ + 9t^4$} \, dt$$

Question 5

Which of the following is the correct formula for the cross product $$\vec{a} 	imes \vec{b} of $$vectors $$\vec{a} = \langle a_1$, a_2$, a_3$ angle$$ and $$\vec{b} = \langle b_1$, b_2$, b_3$ angle$$?

Accepted Answer

$$\langle a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1 \rangle$$

Question 6

Let $$f(x, y) = x^2$y + y^3$. What is the gradient $$
abla f$$ at the point (1$$,2)$$?

Accepted Answer

$$\langle 4, 9 \rangle$$

Question 7

Which of the following best describes the meaning of the divergence $$
abla \cdot \vec{F} of a $$vector field $$\vec{F} in$$ $$\mathbb{R}^3$$?

Accepted Answer

It measures the rate at which the field spreads out from a point.

Question 8

Suppose $$f(x, y) = x^2$ + y^2$ and C$ is the circle x^2$ + y^2$ = 1$$ oriented counterclockwise. What is the value of the line integral $$\int_C 
abla f \cdot d\vec{r}$$?

Accepted Answer

0

Question 9

Which of the following is the correct statement of Green's Theorem for a positively oriented, simple closed curve $$C$$ bounding a region $$D in $$the plane?

Accepted Answer

$$\int_C P \, dx + Q \, dy = \int \int_D \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \, dA$$

Question 10

A weather balloon moves in space along the path $$\vec{r}(t) = \langle \cos t, \sin t, t \rangle$$ for $$t in$$ $$[0, 2\pi]. $$What is the total distance traveled by the balloon?

Accepted Answer

$$\$$int_0$$^{2\pi} \sqrt{(-\sin t)^2$ + (\cos t)^2$ + 1^2$} \, dt$$

Question 11

Which of the following is the correct formula for the surface integral of a scalar function $$f(x, y, z)$$ over a surface $$S$$ parametrized by $$\vec{r}(u, v)$$?

Accepted Answer

$$\iint_S f(x, y, z) \, dS = \iint_D f(\vec{r}(u, v)) \left| \frac{\partial \vec{r}}{\partial u} 	imes \frac{\partial \vec{r}}{\partial v} ight| \, du \, dv$$

Question 12

Let $$\vec{F}(x, y, z) = \langle y, z, x angle. $$What is $$
abla 	imes \vec{F}$$?

Accepted Answer

$$\langle 0, 0, 0 \rangle$$

Question 13

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

Accepted Answer

$$\iint_S 
abla 	imes \vec{F} \cdot d\vec{S} = \int_C \vec{F} \cdot d\vec{r}$$

Question 14

A medical imaging device scans a region $$D in $$the $$xy-$$plane, where $$D is $$the disk $$x^2$$ + $$y^2$$ \leq 4$$. If the density at each point is f(x$$, y) = $$x^2$$ + $$y^2$$, what is the total mass detected by the device?

Accepted Answer

$$\$$int_0$$^{2\pi} \$$int_0$$^2 r^2$ r \, dr \, d	heta$$

Question 15

Which of the following is the correct formula for the directional derivative of $$f(x, y) at$$ $$(x_0$, y_0$) in $$the direction of the unit vector $$\vec{u}$$?

Accepted Answer

$$
abla f(x_0$, y_0$) \cdot \vec{u}$$

Question 16

Which of the following is the correct statement of the Divergence Theorem for a vector field $$\vec{F} on a $$region $$E$$ with boundary surface $$S$$?

Accepted Answer

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

Question 17

A chemical reaction occurs in a region $$E$$ bounded by $$x^2$$ + $$y^2$$ + $$z^2$$ \leq 1$$. If the concentration at each point is f(x$$, y, z) = $$x^2$$ + $$y^2$$ + $$z^2$$, what is the total amount of chemical present in $$E$$?

Accepted Answer

$$\$$int_0$$^{2\pi} \$$int_0$$^{\pi} \$$int_0$$^1 r^2$ r^2$ \sin \phi \, dr \, d\phi \, d	heta$$

Question 18

Which of the following is the correct formula for the line integral of a vector field $$\vec{F}$$ along a curve $$C$$ parametrized by $$\vec{r}(t)$$, $$a \leq t \leq b$$?

Accepted Answer

$$\int_C \vec{F} \cdot d\vec{r} = \int_a^b \vec{F}(\vec{r}(t)) \cdot \vec{r}'(t) \, dt$$

Question 19

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 the value of the triple integral $$\iiint_E f(x, y, z) \, dV$$?

Accepted Answer

9

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