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Exam 17: Vector Calculus

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Question 1

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Find the potential function f for the field F. -

A)
B)
C)
D)

E) None of the above
F) All of the above

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Question 2

Multiple Choice

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Evaluate the work done between point 1 and point 2 for the conservative field F. -

A) W = - 1
B) W = + + - 1
C) W =
D) W = 0

E) A) and B)
F) B) and C)

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Question 3

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Find the mass of the wire that lies along the curve r and has density δ. -r(t) = 7i + ( 9 - 4t) j + 3tk, 0 $\le$ t $\le$ 2 $\pi$ ; = 5(1 + sin 7t)

Evaluate the surface integral of the function g over the surface S. -G(x, y, z) = x² y² z² ; S is the surface of the rectangular prism formed from the planes x = ± 2, y = ± 2, and z = ± 1

Evaluate the work done between point 1 and point 2 for the conservative field F. -F = (y + z) i + xj + xk; (0, 0, 0) , ( 3, 10, 7)

Sketch the vector field in the plane along with its horizontal and vertical components at a representative assortment of points on the circle x² + y² = 4. -F = i + j

Find the surface area of the surface S. -S is the intersection of the plane 3x + 4y + 12z = 7 and the cylinder with sides y = 4 and y = 8 - 4 .

Use Stokes' Theorem to calculate the circulation of the field F around the curve C in the indicated direction. -F = 2yi + 3xj + 6 k ; C: the portion of the plane 3x + 3y + 5z = 6 in the first quadrant

Using the Divergence Theorem, find the outward flux of F across the boundary of the region D. -F = 6 i + 6 j + 6 k ; D: the thick sphere 4 $\le$ + + $\le$ 16

Using Green's Theorem, compute the counterclockwise circulation of F around the closed curve C. -F = - i + j; C is the region defined by the polar coordinate inequalities 8 $\le$ r $\le$ 9 and

Parametrize the surface S. -S is the portion of the cylinder + = 16 that lies between z = 2 and z = 7.

Parametrize the surface S. -S is the portion of the cone + = that lies between z = 1 and z = 9.

Calculate the circulation of the field F around the closed curve C. -F = xyi + 5j , curve C is r(t) = 3 cos ti + 3 sin tj, 0 $\le$ t $\le$ 2 $\pi$

Find the potential function f for the field F. -

Find the gradient field F of the function f. -f(x, y, z) = ln ( + + )

Using the Divergence Theorem, find the outward flux of F across the boundary of the region D. -F = x i + y j + z k ; D: the thick cylinder 1 $\le$ + $\le$ 3 ,

Using Green's Theorem, compute the counterclockwise circulation of F around the closed curve C. -F = xyi + xj; C is the triangle with vertices at (0, 0) , ( 7, 0) , and (0, 7)

Find the flux of the vector field F across the surface S in the indicated direction. -F(x, y, z) = 2xi + 2yj + 2k , S is the surface cut from the bottom of the paraboloid z = + by the plane direction is outward

Parametrize the surface S. -S is the lower portion of the sphere + + = 25 cut by the cone z = .

Calculate the flux of the field F across the closed plane curve C. - the curve C is the closed counterclockwise path formed from the semicircle 0 ≤ t ≤ π, and the straight line segment from (-4, 0) to ( 4, 0)