Question 1

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

Accepted Answer

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

Question 2

Find the curl of the vector field F.

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

Question 3

Let

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The answer of Let  ...

Question 4

Find the curl of the vector field.

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

Question 5

Find the curl of  .

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The answer of Find the curl of  ....

Question 6

Use Green's Theorem and/or a computer algebra system to evaluate  where C is the circle  with counterclockwise orientation.

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The answer of Use Green's Theorem and/or a computer algebra...

Question 7

Find the divergence of the vector field.

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

Question 8

Find the div F if  .

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The answer of Find the div F if  ....

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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The answer of Use Green's Theorem to evaluate the line...

Question 10

Let

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The answer of Let  ...

Question 11

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

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The answer of Find (a) the divergence and (b) the...

Question 12

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...

Question 13

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

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The answer of Let F be a vector field. Determine...

Question 14

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

Question 15

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

Accepted Answer

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

Question 16

Find the curl of the vector field.

Accepted Answer

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

Question 17

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

The answer of Use Green's Theorem to evaluate the line...

Question 18

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

Accepted Answer

The answer of Let F be a vector field. Determine...

Question 19

Find the curl of the vector field.

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

Question 20

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,  ).

Accepted Answer

The centroid of any region with a simple closed path C is given by:
(11eaa3e5_54c0_c868_9f8f_db017abf25c5_TB5971_11)
where A is the area of the region and (x,y) represents the coordinates of any point within the region.

For the triangle with vertices (0, 0), (11eaa3e5_54c0_ef79_9f8f_075812167716_TB5971_11, 0) and (0, 11eaa3e5_54c0_ef7a_9f8f_d1f46e21c0b9_TB5971_11), we can use the formula:

Centroid_x = (x1 + x2 + x3)/3
Centroid_y = (y1 + y2 + y3)/3

where (x1,y1), (x2,y2), and (x3,y3) are the coordinates of the vertices.

Plugging in the values, we get:
Centroid_x = (0 + 11eaa3e5_54c0_ef79_9f8f_075812167716_TB5971_11 + 0)/3 = 11eaa3e5_54c0_ef7e_9f8f_b52619838739_TB5971_11
Centroid_y = (0 + 0 + 11eaa3e5_54c0_ef7a_9f8f_d1f46e21c0b9_TB5971_11)/3 = 11eaa3e5_54c0_ef7f_9f8f_ed782f077129_TB5971_11

Therefore, the centroid of the triangle is (11eaa3e5_54c0_ef7e_9f8f_b52619838739_TB5971_11, 11eaa3e5_54c0_ef7f_9f8f_ed782f077129_TB5971_11), which is closest to choice C: 11eaa3e5_54c0_ef7d_9f8f_677bc5711d64_TB5971_11.

Quiz 16: Vector Calculus

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

Find the curl of the vector field F.

Let

Find the curl of the vector field.

Find the curl of .

Use Green's Theorem and/or a computer algebra system to evaluate where C is the circle with counterclockwise orientation.

Find the divergence of the vector field.

Find the div F if .

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 .

Let

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

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 F be a vector field. Determine whether the expression is meaningful. If so, state whether the expression represents a scalar field or a vector field. curl (div F)

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

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

Find the curl of the vector field.

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

Find the curl of the vector 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, ) .