Exam 17: Vector Calculus

Evaluate the integral where R is the region + + $\le$ 25 and

The local bases in cylindrical coordinates (r , $\theta$ , z) are given by = cos( $\theta$ ) i + sin( $\theta$ ) j , = - sin( $\theta$ ) i + cos( $\theta$ ) j, and = k. Express the acceleration a of a moving particle in space in terms of the local bases above.

If × F = 0 at every point of a simply connected open region D in 3-space, evaluate for any piecewise smooth closed curve in D.

Evaluate the integral - dx counterclockwise around the closed curve formed by y = x³ and y = x, between the points (0, 0) and (1, 1).

A certain region R in 3-space has volume 5 cubic units and centroid at the point (2, -3, 4). Find the flux of out of R across its boundary.

Use Green's Theorem to compute + xy) dx + ( + xy) dy counterclockwise around the rectangle having vertices (± 1, 1) and (± 1, 2).

Compute $\textbf{ div F }$ for $\textbf{ F }$ = sin 2x, cos 2y, tan 2z .

Use the Divergence Theorem to evaluate the surface integral where S is the part of the cone below z = 2, and is the unit normal to S with positive z-component. (An additional surface must be introduced to enclose a volume.)

Let be a scalar field and F be a vector field, both assumed to be sufficiently smooth. Which of the following expressions is meaningless?

Find the flux of G = (x + 2zy) i + (y - ) j + z k outward through the sphere

Find the flux of F = x i + 2y j out of the circular disk of radius 2 centred at (3, -5).

Let B be a constant vector and let G(r) = (B × r) × r be a vector potential of the solenoidal vector field F. Find F.

The local bases in cylindrical coordinates (r , $\theta$ , z) are given by = cos( $\theta$ ) i + sin( $\theta$ ) j, = - sin( $\theta$ ) i + cos( $\theta$ ) j, and = k.Express the velocity v of a moving particle in space in terms of the local bases above.

Calculate the curl of the vector field V = x sin y i + cos y j + xy k.

Evaluate the integral of over the portion of the ellipse in the first quadrant, traversed in the counterclockwise direction.

Evaluate the line integral where C is the circle oriented clockwise as seen from high on the z-axis.

Use the Divergence Theorem to find the outward flux of F = across the boundary of the region

Let F = (z - y) i + (x - z) j + (y - x) k. Compute the work done by the force F in moving an object along the curve of intersection of the cylinder with the plane The orientation of the curve is consistent with the upward normal on the plane.

Use Green's Theorem to compute the integral counterclockwise around the square with vertices at (4, 2), (4, 5), (7, 5), and (7, 2).