Q: Given the vector field $$\vec{F}(x, y, z) = (x, y, z)$$, what is the flux through the surface $$S$$ defined by $$x^2$$ + $$y^2$$ + $$z^2 = 1$ (the unit sphere) using the Divergence Theorem?

$$3 \times \frac{4}{3}\pi = 4\pi$$

Q: Suppose you are converting the rectangular coordinates $$(x, y) to $$polar coordinates. Which of the following is correct?

$$x = r \cos \theta, y = r \sin \theta$$

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

$$\int_C \vec{F} \cdot d\vec{r} = \int_S (\nabla \times \vec{F}) \cdot d\vec{A}$$

Q: Which of the following is the correct formula for the divergence of a vector field $$\vec{F} = (F_1$, F_2$, F_3$) in $$rectangular coordinates?

$$\nabla \cdot \vec{F} = \frac{\partial F_1$}{\partial x} + \frac{\partial F_2$}{\partial y} + \frac{\partial F_3$}{\partial z}$$

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

$$\int_S \vec{F} \cdot d\vec{A} = \int_V (\nabla \cdot \vec{F}) \, dV$$

Q: Which of the following is the correct formula for the surface area element $$dA in $$spherical coordinates?

$$dA = r^2$ \sin \phi \, d\theta \, d\phi$$

Question 1

Which of the following is the correct scalar formula for the flux of a vector field $$\vec{F}$$ through a surface $$S$$?

Accepted Answer

$$\int_S \vec{F} \cdot \vec{n} \, dA$$

Question 2

Given the vector field $$\vec{F}(x, y, z) = (x, y, z)$$, what is the flux through the surface $$S$$ defined by $$x^2$$ + $$y^2$$ + $$z^2 = 1$ (the unit sphere) using the Divergence Theorem?

Accepted Answer

$$3 	imes \frac{4}{3}\pi = 4\pi$$

Question 3

Which of the following is the correct expression for the line integral of a vector field $$\vec{F}$$ along a curve $$C$$?

Accepted Answer

$$\int_C \vec{F} \cdot d\vec{r}$$

Question 4

In cylindrical coordinates, which of the following correctly represents the variables?

Accepted Answer

$$r, 	heta, z$$

Question 5

Suppose you are converting the rectangular coordinates $$(x, y) to $$polar coordinates. Which of the following is correct?

Accepted Answer

$$x = r \cos 	heta, y = r \sin 	heta$$

Question 6

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

Accepted Answer

$$\int_C \vec{F} \cdot d\vec{r} = \int_S (
abla 	imes \vec{F}) \cdot d\vec{A}$$

Question 7

If a vector field $$\vec{F} is $$conservative, which of the following is true for any closed curve $$C$$?

Accepted Answer

$$\int_C \vec{F} \cdot d\vec{r} = 0$$

Question 8

Which of the following is the correct formula for the divergence of a vector field $$\vec{F} = (F_1$, F_2$, F_3$) in $$rectangular coordinates?

Accepted Answer

$$
abla \cdot \vec{F} = \frac{\partial F_1$}{\partial x} + \frac{\partial F_2$}{\partial y} + \frac{\partial F_3$}{\partial z}$$

Question 9

A fluid flows through a surface $$S$$ with velocity field $$\vec{F}. $$Which integral gives the total rate at which fluid passes through $$S$$?

Accepted Answer

$$\int_S \vec{F} \cdot \vec{n} \, dA$$

Question 10

Which of the following is the correct formula for the curl of a vector field $$\vec{F} = (F_1$, F_2$, F_3$)$$?

Accepted Answer

Question 11

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

Accepted Answer

$$\int_S \vec{F} \cdot d\vec{A} = \int_V (
abla \cdot \vec{F}) \, dV$$

Question 12

If $$x = r \cos 	heta$$ and $$y = r \sin 	heta$$, what is the Jacobian determinant for the transformation from rectangular to polar coordinates?

Accepted Answer

$$r$$

Question 13

Which of the following is the correct formula for the surface area element $$dA in $$spherical coordinates?

Accepted Answer

$$dA = r^2$ \sin \phi \, d	heta \, d\phi$$

Question 14

A vector field $$\vec{F} is $$given by $$\vec{F} = 
abla f$$ for some scalar function $$f. $$Which property does $$\vec{F}$$ necessarily have?

Accepted Answer

$$\vec{F} is $$conservative

Question 15

Which of the following is the correct formula for the volume element $$dV in $$cylindrical coordinates?

Accepted Answer

$$dV = r \, dr \, d	heta \, dz$$

Question 16

Suppose you are given a vector field $$\vec{F}$$ and a closed surface $$S$$ enclosing a volume $$V. $$Which theorem allows you to compute the flux of $$\vec{F}$$ through $$S by $$integrating the divergence of $$\vec{F}$$ over $$V$$?

Accepted Answer

Divergence Theorem

Question 17

Which of the following is the correct statement of Green's Theorem for a region $$D$$ bounded by a positively oriented curve $$C$$?

Accepted Answer

Vector Calculus: Integrals, Coordinate Systems, and Theorems

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