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Exam 17: Vector Calculus

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Question 21

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In cylindrical coordinates, find f for f = sin( $\theta$ ) + .

A) 2r sin( $\theta$ ) + r cos( $\theta$ ) + 2z k
B) 2r sin( $\theta$ ) + cos( $\theta$ ) + 2z k
C) 2r cos( $\theta$ ) + r sin( $\theta$ ) + 2z k
D) 2r cos( $\theta$ ) + sin( $\theta$ ) + 2z k
E) r sin( $\theta$ ) + cos( $\theta$ ) + 2z k

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

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Question 22

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In spherical coordinates, find f for f = sin( $\theta$ ) cos( $\theta$ ) .

A) 3 sin( $\theta$ ) cos( $\theta$ ) + cos( $\theta$ ) cos( $\theta$ ) - sin( $\theta$ )
B) 3 sin( $\theta$ ) cos( $\theta$ ) + cos( $\theta$ ) cos( $\theta$ ) - sin( $\theta$ ) sin( $\theta$ )
C)
D) 3 sin( $\theta$ ) cos( $\theta$ ) - cos( $\theta$ ) cos( $\theta$ ) + sin( $\theta$ )
E) 3 sin( $\theta$ ) cos( $\theta$ ) - cos( $\theta$ ) cos( $\theta$ ) - sin( $\theta$ )

F) None of the above
G) A) and C)

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Question 23

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In cylindrical coordinates, find . F for F = + rz cos( $\theta$ ) + rz sin( $\theta$ ) k.

Use Green's theorem in the plane to find the x-coordinate of the centroid of a regular plane region R (with areaA) enclosed by a positively oriented, piecewise smooth, simple closed curve C .

Let F = - i + j and let C be the boundary of circle + = 9 oriented counterclockwise. Use Green's Theorem to evaluate

Using spherical polar coordinates, find × F for F = sin( $\theta$ ) + sin( $\theta$ ) + cos( $\theta$ ) .

If r = x i + y j + z k and k is a constant vector field in R³, then

Evaluate where S is the first-octant part of the sphere of radius a centred at the origin. (Hint: Even though S is not a closed surface, it is still easiest to use the Divergence Theorem because the integrand in the surface integral is zero on the coordinate planes.)

Use Green's Theorem to compute the integral where C is the triangle formed by the lines y = -x + 1, x = 0 and y = 0, oriented clockwise.

Find all values of the nonzero constant real numbers a, b, and c so that the vector field F = a cos( x + 2y ) cosh (c z) i + b cos ( x + 2y) cosh (c z) j + c sin( x + 2y) sinh(c z) k is both $\textbf{ irrotational }$ and $\textbf{ solenoidal }$ .

Use Stokes's Theorem to evaluate the integral where C is the curve of intersection of the sphere and the plane oriented counterclockwise as seen from high on the z-axis.

If C is the positively oriented boundary of a plane region R having area 3 units and centroid at the point (12, 6) , evaluate (i) (ii) dx + 3xy dy

Use Stokes's Theorem to evaluate the line integral where C is the triangle with vertices (0, 0, 1) , (0, 1, 1) and (1, 0, 0) with counterclockwise orientation as seen from high on the z-axis.

Let and F be sufficiently smooth scalar and vector fields, respectively.Express the well-known identity . (11ee7bad_9f85_f900_ae82_29e1b84eee54_TB9661_11 F ) = (11ee7bad_7817_372f_ae82_a36163e56c30_TB9661_11 11ee7bad_9f85_f900_ae82_29e1b84eee54_TB9661_11 ) . F + 11ee7bad_9f85_f900_ae82_29e1b84eee54_TB9661_11 (11ee7bad_7817_372f_ae82_a36163e56c30_TB9661_11. F) using the notations grad , div or curl.

Evaluate clockwise around the triangle with vertices (0, 0) , (3, 0) , and (3, 3) .

Define the curl of a vector field F.

Find the flux of i - xy j +3z k out of the solid region bounded by the parabolic cylinder and the planes , and

Evaluate the integral ( ) - 2y) dx + (3x - ysin( ) ) dy counterclockwise around the triangle in the xy-plane having vertices (0, 0) , (2, 2) , and (2, 0) .

Let F be a smooth vector field in 3-space satisfying the condition Find the flux of curl F upward through the part of the lying above the xy-plane.

A vector field F is called $\textbf{ solenoidal }$ in a domain D if