Exercise ​

Consider the parametric surface

1. Write the equation for which describes this surface.
2. Find the fundamental vector product of .

Exercise ​

Consider the parametric surface

1. Write the equation for which describes this surface.
2. Find the fundamental vector product of .

Exercise ​

Let be the bounded portion of the paraboloid which is cut off by the plane . Sketch and choose a parametric representation of this surface. Hint: a possible parametric representation is , for a suitable choice of .

Use this to compute the area of .

Exercise ​

Let denote the plane surface whose boundary is the triangle with vertices at , and . Consider the vector field .

Compute the surface integral , where denotes the unit normal to which has positive -component.

Exercise ​

Consider the vector field

Calculate the curl and divergence of and find

Exercise ​

Consider the vector field

Calculate and .

Exercise ​

Let

be a vector field. Let be the portion of the paraboloid with and let be the unit normal to with non-negative -component. Using Stokes' Theorem (to transform the surface integral to a line integral), evaluate .

Exercise ​

Let be the set of such that and . This solid is bounded by a closed surface which is composed of two parts: Let denote the curved top part and let denote the planar part.

Consider the vector field

1. Evaluate the surface integral .
2. Evaluate the multiple integral .
3. Evaluate the surface integral .

Exercise ​

Consider the surface . Find a parametric form for based on polar coordinates , and find the associated fundamental vector product.

Use this to find the surface integral , where is the vector field

and is the unit normal to which has positive -component.

Exercise ​

Consider the the sphere and the vector-field

Calculate , where is the outgoing unit normal on .

Hint: Use Gauss's theorem to rewrite this as a volume integral.

Exercise ​

Let be the part of the surface with , oriented in the positive direction. Find

with .