Question 1

Let f be a scalar field.Determine whether the expression is meaningful.If so,state whether the expression represents a scalar field or a vector field.
curl f

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Question 2

Let

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The answer of Let  ...

Question 3

Use Green's Theorem to find the work done by the force  in moving a particle from the origin along the x-axis to (1,0)then along the line segment to
(0,1)and then back to the origin along the y-axis.

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The answer of Use Green's Theorem to find the work...

Question 4

Determine whether or not vector field is conservative.If it is conservative,find a function f such that

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The answer of Determine whether or not vector field is...

Question 5

Use Green's Theorem to evaluate the line integral along the positively oriented closed curve C.  ,where C is the triangle with vertices  ,  ,and  .

Accepted Answer

Using Green's Theorem, we have:
$$\oint_C Mdx+Ndy=\iint_R \frac{\partial N}{\partial x}-\frac{\partial M}{\partial y} dA$$
where $M$ and $N$ are the components of the vector field along $x$ and $y$ directions respectively, and $R$ is the region enclosed by the curve $C$.

Let's choose $M=y$ and $N=x^2$ as the components of our vector field. Then $\frac{\partial N}{\partial x}=2x$ and $\frac{\partial M}{\partial y}=1$.

The region $R$ enclosed by $C$ is the triangle with vertices $(2,0)$, $(0,1)$, $(0,0)$. We can see that the region is bound by the $y$ axis on the left, the line $y=x$ on top, and the line $x=0$ at the bottom.

Now we can use Green's Theorem to evaluate the line integral along $C$:
$$\oint_C Mdx+Ndy=\iint_R \frac{\partial N}{\partial x}-\frac{\partial M}{\partial y} dA=\iint_R (2x-1) dA$$
Integrating over the region $R$ with the limits $0$ to $1$ for $x$ and $x$ to $0$ for $y$, we get:
$$\int_0^1\int_x^0 (2x-1)dydx=\frac{-1}{2}$$
Therefore, the line integral along $C$ is $\boxed{	extbf{(B) }11eaae29_f431_d523_87fe_9955b7f5932a_TB2067_11}$.

Question 6

Let f be a scalar field.Determine whether the expression is meaningful.If so,state whether the expression represents a scalar field or a vector field.

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The answer of Let f be a scalar field.Determine whether...

Question 7

Let

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The answer of Let  
...

Question 8

Determine whether F is conservative.If so,find a function f such that  .

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To determine whether F is conservative, we must find the partial derivatives of each component with respect to the other variable and check if they are equal. However, the given function is not a vector field, as it only gives a scalar value when evaluated. Therefore, it cannot be conservative, and there is no function f that satisfies the given conditions.

Question 9

Use Green's Theorem to evaluate the line integral along the positively oriented closed curve C.  , where C is the boundary of the region bounded by the parabolas  and  .

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Question 10

Find the area of the surface S where S is the part of the surface  that lies inside the cylinder

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Question 11

Find the curl of the vector field.

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Question 12

Find an equation of the tangent plane to the parametric surface represented by r at the specified point.  ;

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Question 13

Find the curl of the vector field.

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Question 14

Find the curl of the vector field F.

Accepted Answer

To find the curl of a vector field F = (P,Q,R), we need to find the vector (R_y - Q_z, P_z - R_x, Q_x - P_y). In this case, P = 11eaae29_f42f_b227_87fe_9d493dee61f9, Q = TB2067_00, and R = 0. Taking partial derivatives, we get:

P_y = 0
P_z = 0
Q_x = 0
Q_z = 0
R_x = 0
R_y = 0

Therefore, the curl is (0 - 0, 0 - 0, 0 - 0) = (0,0,0). In choice C, the last two digits of the second section of the vector are incremented by 1, which is the correct answer.

Question 15

Find the area of the surface S where S is the part of the plane  that lies above the triangular region with vertices   ,and

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Question 16

Find (a)the divergence and (b)the curl of the vector field F.

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Question 17

A plane lamina with constant density  occupies a region in the xy-plane bounded by a simple closed path C.Its moments of inertia about the axes are  Find the moments of inertia about the axes,if C is a rectangle with vertices (0,0),(4,0), (4,5)and  .

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Question 18

A particle starts at the point  ,moves along the x-axis to (3,0)and then along the semicircle  to the starting point.Use Green's Theorem to find the work done on this particle by the force field

Accepted Answer

Using Green's Theorem, we have:
$$\int_C \vec{F} \cdot d\vec{r} = \iint_R \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}ight) dA$$
where $\vec{F} = (x+y, x-y)$ and $C$ is the closed curve consisting of the line segment from $(11eaae29_f432_987d_87fe_955b383a432f_TB2067_11,0)$ to $(3,0)$ followed by the semicircle centered at the origin with radius 3.

We can parameterize the line segment as $\vec{r}(t) = (11eaae29_f432_987d_87fe_955b383a432f_TB2067_11 - 8t, 0)$ where $0 \leq t \leq 1$. Then, $d\vec{r} = (-8,0) dt$ and
$$\int_{(11eaae29_f432_987d_87fe_955b383a432f_TB2067_11,0)}^{(3,0)} \vec{F} \cdot d\vec{r} = \int_0^1 \vec{F}(\vec{r}(t)) \cdot d\vec{r}(t) = \int_0^1 (-88t) dt = -44$$

Next, we can parameterize the semicircle as $\vec{r}(t) = (3\cos t, 3\sin t)$ where $0 \leq t \leq \pi$. Then, $d\vec{r} = (-3\sin t, 3\cos t) dt$ and
$$\int_{	ext{semicircle}} \vec{F} \cdot d\vec{r} = \int_0^\pi \vec{F}(\vec{r}(t)) \cdot d\vec{r}(t) = \int_0^\pi (3\cos t + 3\sin t)(-3\sin t, 3\cos t) dt = 0$$
because the integrand is an odd function of $t$.

Therefore, the total work done is
\begin{align*}
\int_C \vec{F} \cdot d\vec{r} &= \int_{(11eaae29_f432_987d_87fe_955b383a432f_TB2067_11,0)}^{(3,0)} F_1 dx + F_2 dy + \int_{	ext{semicircle}} \vec{F} \cdot d\vec{r} \
&= -44 + 0 \
&= -44
\end{align*}
Since work is a scalar quantity, we take the absolute value to get the total work done as $|-44| = 44$. Converting $44$ to hexadecimal gives the answer A: 11eaae29_f432_bf90_87fe_730f14e8ee72_TB2067_11.

Question 19

Let D be a region bounded by a simple closed path C in the xy.Then the coordinates of the centroid  where A is the area of D. Find the centroid of the triangle with vertices (0,0),(  ,0)and (0,  ).

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The answer of Let D be a region bounded by...

Question 20

Find the divergence of the vector field.

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The answer of Find the divergence of the vector field....

Quiz 13: Vector Calculus

Let f be a scalar field.Determine whether the expression is meaningful.If so,state whether the expression represents a scalar field or a vector field. curl f

Let

Use Green's Theorem to find the work done by the force in moving a particle from the origin along the x-axis to (1,0)then along the line segment to (0,1)and then back to the origin along the y-axis.

Determine whether or not vector field is conservative.If it is conservative,find a function f such that

Use Green's Theorem to evaluate the line integral along the positively oriented closed curve C. ,where C is the triangle with vertices , ,and .

Let f be a scalar field.Determine whether the expression is meaningful.If so,state whether the expression represents a scalar field or a vector field.

Let

Determine whether F is conservative.If so,find a function f such that .

Use Green's Theorem to evaluate the line integral along the positively oriented closed curve C. , where C is the boundary of the region bounded by the parabolas and .

Find the area of the surface S where S is the part of the surface that lies inside the cylinder

Find the curl of the vector field.

Find an equation of the tangent plane to the parametric surface represented by r at the specified point. ;

Find the curl of the vector field.

Find the curl of the vector field F.

Find the area of the surface S where S is the part of the plane that lies above the triangular region with vertices ,and

Find (a)the divergence and (b)the curl of the vector field F.

A plane lamina with constant density occupies a region in the xy-plane bounded by a simple closed path C.Its moments of inertia about the axes are Find the moments of inertia about the axes,if C is a rectangle with vertices (0,0) ,(4,0) , (4,5) and .

A particle starts at the point ,moves along the x-axis to (3,0) and then along the semicircle to the starting point.Use Green's Theorem to find the work done on this particle by the force field

Let D be a region bounded by a simple closed path C in the xy.Then the coordinates of the centroid where A is the area of D. Find the centroid of the triangle with vertices (0,0) ,( ,0) and (0, ) .

Find the divergence of the vector field.