Solved: (a) What is an oriented surface? Give an example of a nonorientable surface. (b)

What is a vector field? Give three examples that have physical meaning.

(a) What is a conservative vector field? (b) What is a potential function?

(a) Write the definition of the line integral of a scalar function along a smooth curve with respect to arc length. (b) How do you evaluate such a line integral? (c) Write expressions for the mass and center of mass of a thin wire shaped like a curve if the wire has linear density function . (d) Write the definitions of the line integrals along of a scalar function with respect to , , and . (e) How do you evaluate these line integrals?

(a) Define the line integral of a vector field along a smooth curve given by a vector function . (b) If is a force field, what does this line integral represent? (c) If , what is the connection between the line integral of and the line integrals of the component functions , , and ?

State the Fundamental Theorem for Line Integrals.

(a) What does it mean to say that is independent of path? (b) If you know that is independent of path, what can you say about ?

State Greens Theorem.

Write expressions for the area enclosed by a curve in terms of line integrals around .

Suppose is a vector field on . (a) Define curl . (b) Define div . F (c) If is a velocity field in fluid flow, what are the physical interpretations of curl and div ?

If , how do you test to determine whether is conservative? What if is a vector field on ?

(a) What is a parametric surface? What are its grid curves? (b) Write an expression for the area of a parametric surface. (c) What is the area of a surface given by an equation ?

(a) Write the definition of the surface integral of a scalar function over a surface . (b) How do you evaluate such an integral if is a parametric surface given by a vector function ? (c) What if is given by an equation ? (d) If a thin sheet has the shape of a surface , and the density at is , write expressions for the mass and center of mass of the sheet.

(a) What is an oriented surface? Give an example of a nonorientable surface. (b) Define the surface integral (or flux) of a vector field F over an oriented surface S with unit normal vector n. (c) How do you evaluate such an integral if S is a parametric surface given by a vector function ? (d) What if S is given by an equation ?

State Stokes Theorem.

State the Divergence Theorem.

In what ways are the Fundamental Theorem for Line Integrals, Greens Theorem, Stokes Theorem, and the Divergence Theorem similar?

Use Greens Theorem to evaluate , where is the circle with counterclockwise orientation.

Find curl and div if

Show that there is no vector field such that curl G 2x i  3yz j xz2 k G

Show that, under conditions to be stated on the vector fields and , urlF G F div G G div F  G F F GGF

If is any piecewise-smooth simple closed plane curve and and are differentiable functions, show that C fx dx  ty dy 0

If and are twice differentiable functions, show that 2  ft f 2 t  t2 f  2f t f

If is a harmonic function, that is, , show that the line integral is independent of path in any simple region D

a) Sketch the curve with parametric equations (b) Find .

Find the area of the part of the surface that lies above the triangle with vertices , , and .

(a) Find an equation of the tangent plane at the point to the parametric surface S given by ru, v v 0 u 3 , 3 v 3 2 i (b) Use a computer to graph the surface and the tangent plane found in part (a). (c) Set up, but do not evaluate, an integral for the surface area of . (d) If find correct to four decimal places.

Evaluate the surface integral.

Evaluate the surface integral.

Evaluate the surface integral.

Evaluate the surface integral.

Verify that Stokes Theorem is true for the vector field where is the part of the paraboloid that lies above the -plane and has upward orientation.

Use Stokes Theorem to evaluate , where , is the part of the sphere that lies above the plane , and is oriented upward.

Use Stokes Theorem to evaluate , where and is the triangle with vertices , , and , oriented counterclockwise as viewed from above.

Use the Divergence Theorem to calculate the surface integral , where and is the surface of the solid bounded by the cylinder and the planes and .

Verify that the Divergence Theorem is true for the vector field where is the unit ball .

Compute the outward flux of through the ellipsoid . 4x 2  9y 2  6z2 36 Fx, y, z x i  y j  z k x 2  y 2  z2  32

Let Evaluate , where is the curve with initial point and terminal point shown in the figure.

Let Evaluate , where is shown in the figure.

Find , where and is the outwardly oriented surface shown in the figure (the boundary surface of a cube with a unit corner cube removed).

If the components of have continuous second partial derivatives and is the boundary surface of a simple solid region, show that . x