Question 1

Suppose that F is an inverse square force field,that is,  where  Find the work done by F in moving an object from a point  along a path to a point  in terms of the distances  and  from these points to the origin.

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The answer of Suppose that F is an inverse square...

Question 2

The flow lines (or streamlines)of a vector field are the paths followed by a particle whose velocity field is the given vector field.Thus,the vectors in a vector field are tangent to the flow lines.The flow lines of the vector field  satisfy the differential equations  and  Solve these differential equations to find the equations of the family of flow lines.

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The answer of The flow lines (or streamlines)of a vector...

Question 3

Find the exact mass of a thin wire in the shape of the helix  if the density is 5.
A) 
B) 
C) 
D) 
E)

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The answer of Find the exact mass of a thin...

Question 4

Find the gradient vector field of
A) 
B) 
C) 
D) 
E) None of these

Accepted Answer

The answer of Find the gradient vector field of
A) 
B)...

Question 5

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

Accepted Answer

The answer of Determine whether or not F is a...

Question 6

A particle is moving in a velocity field  At time t = 1 the particle is located at the point (1,5,5).
a)What is the velocity of the particle at t = 1?
b)What is the approximate location of the particle at t = 1.01?

Accepted Answer

The answer of A particle is moving in a velocity...

Question 7

A thin wire is bent into the shape of a semicircle  If the linear density is 4 ,find the exact mass of the wire.
A) 
B) 
C) 
D) 
E)

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The answer of A thin wire is bent into the...

Question 8

Find a function f such that  ,and use it to evaluate  along the given curve C.

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The answer of Find a function f such that ...

Question 9

Find the work done by the force field  on a particle that moves along the parabola 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Find the work done by the force...

Question 10

Find the gradient vector field of

Accepted Answer

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

Question 11

Evaluate  where C is the right half of the circle 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The given expression represents the tangent of an angle whose vertex lies on the circle and whose rays intersect the x-axis at the given point. Since C is the right half of the circle, the angle whose vertex lies on C and whose rays intersect the x-axis at the given point is in the second quadrant. Thus, we can use the reference angle in the first quadrant, which is complement to the angle in the second quadrant. The reference angle is the angle whose tangent is 11eaae29_f436_4231_87fe_51bd301065d6_TB2067_11_TB2234_11.

Let x be the reference angle in the first quadrant.

tan(x) = 11eaae29_f436_4231_87fe_51bd301065d6_TB2067_11_TB2234_11 
tan(x) = sin(x)/cos(x)

Let r be the radius of the circle.

sin(x) = sqrt(r^2 - (distance from point to x-axis)^2) 
cos(x) = distance from point to y-axis

The distance from the point to x-axis is r/2 and the distance from the point to y-axis is sqrt(3)r/2.

sin(x) = sqrt(r^2 - (r/2)^2) = sqrt(3)r/2 
cos(x) = sqrt(3)r/2

tan(x) = (sqrt(3)r/2)/(sqrt(3)r/2) = 1

Thus, the reference angle is 45 degrees or pi/4 radians.

Since the angle we want is in the second quadrant, its measure is pi - pi/4 = 3pi/4 radians.

tan(3pi/4) = -1, since the tangent is negative in the second quadrant.

Therefore, evaluating the expression gives:

11eaae29_f436_4230_87fe_a9563345008b_TB2067_11 = -1

The only choice that gives a hexadecimal value of -1 is C: 
11eaae29_f436_9054_87fe_b9c80fc811cf_TB2067_11

Question 12

Let  where  . Which of the following equations does the line segment from  to  satisfy?
A) 
B) 
C) none of these

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The answer of Let  where  . Which of...

Question 13

Which plot illustrates the vector field 
A) 
B) 
C) 
D)

Accepted Answer

To determine the plot that illustrates the vector field, we need to look at the direction of the vectors in the field. The vector field is represented by a set of vectors, each with a magnitude and a direction. In this case, the direction of the vectors can be determined by looking at the orientation of the black lines on each plot.
Plot A has vectors that are oriented horizontally, whereas the vector field is oriented in a northwest direction. Plot B has vectors that are oriented vertically, which is also not consistent with the vector field. Plot D has vectors that are oriented in a northeast direction, which is close but not exactly the same as the direction of the vector field.
Plot C, on the other hand, has vectors that are oriented in a northwest direction, which is consistent with the direction of the vector field. Therefore, the best choice for the plot that illustrates the vector field is C.

Question 14

Find the work done by the force field  in moving an object along an arch of the cycloid

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The answer of Find the work done by the force...

Question 15

Evaluate the line integral over the given curve C.  ,where C is the line segment joining (-2,-1)to (4,5)

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The answer of Evaluate the line integral over the given...

Question 16

Evaluate the line integral over the given curve C.  ;  , 
A) 
B)  
C)  
D)

Accepted Answer

The answer of Evaluate the line integral over the given...

Quiz 13: Vector Calculus

Suppose that F is an inverse square force field,that is, where Find the work done by F in moving an object from a point along a path to a point in terms of the distances and from these points to the origin.

Find the exact mass of a thin wire in the shape of the helix if the density is 5.

Find the gradient vector field of

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

A particle is moving in a velocity field At time t = 1 the particle is located at the point (1,5,5). a)What is the velocity of the particle at t = 1? b)What is the approximate location of the particle at t = 1.01?

A thin wire is bent into the shape of a semicircle If the linear density is 4 ,find the exact mass of the wire.

Find a function f such that ,and use it to evaluate along the given curve C.

Find the work done by the force field on a particle that moves along the parabola

Find the gradient vector field of

Evaluate where C is the right half of the circle

Let where . Which of the following equations does the line segment from to satisfy?

Which plot illustrates the vector field

Find the work done by the force field in moving an object along an arch of the cycloid

Evaluate the line integral over the given curve C. ,where C is the line segment joining (-2,-1)to (4,5)

Evaluate the line integral over the given curve C. ; ,