Question 1

Evaluate  where  and C is any positively oriented simple closed curve enclosing the origin.
A) 0
B) 
C) 
D)

Accepted Answer

Since the closed curve C encloses the origin, we can apply Cauchy's Integral Theorem to conclude that the integral is equal to 0. This theorem states that if f(z) is analytic in and on a simply connected region R and C is any closed curve in R, then the integral of f(z) around C is equal to 0.

Question 2

Determine whether  is conservative, and if so, find a potential function  of the vector field.
A)  is conservative with 
B)  is conservative with 
C)  is conservative with 
D)  is not conservative

Accepted Answer

The answer of Determine whether  is conservative, and if...

Question 3

Set up a line integral to compute the area of the region bounded by  and

Accepted Answer

The answer of Set up a line integral to compute...

Question 4

Find the curl of  .
A) 
B) 
C) 
D)

Accepted Answer

The answer of Find the curl of  .
A) 
B)...

Question 5

Find a potential function and evaluate the integral  where C runs from  to

Accepted Answer

The answer of Find a potential function and evaluate the...

Question 6

Use a line integral to compute the area of the region bounded by the ellipse  .
A) 
B) 
C) 
D)

Accepted Answer

We can parameterize the ellipse as 
$$\vec{r}(t) = \langle a\cos(t), b\sin(t)angle,\quad 0\leq t \leq 2\pi$$
where $a$ and $b$ are the major and minor axes, respectively. In this case, $a=3$ and $b=2$, so 
$$\vec{r}(t) = \langle 3\cos(t), 2\sin(t) angle,\quad 0\leq t \leq 2\pi$$
The area of the ellipse can be found using a line integral by computing
$$A = \frac{1}{2}\oint_{C} xdy-ydx$$
where $C$ is the boundary of the ellipse, which is given by $\vec{r}(t)$. Evaluating this integral yields
\begin{align*}
A &= \frac{1}{2}\int_{0}^{2\pi} \left( 3\cos(t) \cdot 2\cos(t) - 2\sin(t)\cdot(-3\sin(t)) ight) dt \
&= \frac{1}{2}\int_{0}^{2\pi} 6\cos^2(t) + 6\sin^2(t) dt \
&= \frac{1}{2}\int_{0}^{2\pi} 6 dt \
&= 3\pi
\end{align*}
Therefore, the correct answer is choice C.

Question 7

Evaluate  where  and C is the line segment from  to 
A) 
B) 
C)  
D)

Accepted Answer

The answer of Evaluate  where  and C is...

Question 8

Use Green's Theorem to evaluate  where C is the rectangle from  to  to  to  .
A) -1080
B) -135
C) -45
D) 0

Accepted Answer

The answer of Use Green's Theorem to evaluate  where...

Question 9

Determine whether  is conservative, and if so, find a potential function  of the vector field.
A)  is conservative with 
B)  is conservative with 
C)  is conservative with 
D)  is not conservative

Accepted Answer

The answer of Determine whether  is conservative, and if...

Question 10

Find the divergence of  .
A) 
B) 
C) 
D)

Accepted Answer

The answer of Find the divergence of  .
A) 
B)...

Question 11

Use Green's Theorem to evaluate  where  and C is formed by  and  Assume the curve is oriented positively.
A) 
B) 
C) 
D)

Accepted Answer

The answer of Use Green's Theorem to evaluate  where...

Question 12

Find the curl of  .
A) 
B) 
C) 
D)

Accepted Answer

The answer of Find the curl of  .
A) 
B)...

Question 13

Use Green's Theorem to evaluate  where C is curve bounded by  and 
A) 
B) 64
C) 
D) 0

Accepted Answer

The answer of Use Green's Theorem to evaluate  where...

Question 14

Evaluate  for  and C is the quarter circle from  to  .
A) 0
B) 
C) 
D)

Accepted Answer

The answer of Evaluate  for  and C is...

Question 15

Evaluate  for  and C is the top half-circle  from  to  .
A) 
B) 
C) 0
D)

Accepted Answer

The answer of Evaluate  for  and C is...

Question 16

Find the curl of  .
A) 
B) 
C) 
D)

Accepted Answer

The answer of Find the curl of  .
A) 
B)...

Question 17

Use Green's Theorem to evaluate  where C is curve bounded by  and  oriented clockwise.
A) 
B) 
C) 
D) 0

Accepted Answer

The answer of Use Green's Theorem to evaluate  where...

Question 18

Show that the line integral  is independent of the path.

Accepted Answer

The answer of Show that the line integral  is...

Question 19

Find the divergence of  .
A) 0
B) 
C) 
D)

Accepted Answer

The divergence of a scalar field is always zero.

Question 20

Find the divergence of  .
A) 
B) 
C) 
D)

Accepted Answer

The answer of Find the divergence of  .
A) 
B)...

Quiz 16: Vector Calculus

Evaluate where and C is any positively oriented simple closed curve enclosing the origin.

Determine whether is conservative, and if so, find a potential function of the vector field.

Set up a line integral to compute the area of the region bounded by and

Find the curl of .

Find a potential function and evaluate the integral where C runs from to

Use a line integral to compute the area of the region bounded by the ellipse .

Evaluate where and C is the line segment from to

Use Green's Theorem to evaluate where C is the rectangle from to to to .

Determine whether is conservative, and if so, find a potential function of the vector field.

Find the divergence of .

Use Green's Theorem to evaluate where and C is formed by and Assume the curve is oriented positively.

Find the curl of .

Use Green's Theorem to evaluate where C is curve bounded by and

Evaluate for and C is the quarter circle from to .

Evaluate for and C is the top half-circle from to .

Find the curl of .

Use Green's Theorem to evaluate where C is curve bounded by and oriented clockwise.

Show that the line integral is independent of the path.

Find the divergence of .

Find the divergence of .