Multivariable Calculus : Triple Integration of Surface

Example Questions

Example Question #1 : Divergence, Gradient, & Curl

Calculate the curl for the following vector field.

Explanation:

In order to calculate the curl, we need to recall the formula.

where , and  correspond to the components of a given vector field:

Now lets apply this to out situation.

Thus the curl is

Example Question #2 : Divergence, Gradient, & Curl

Compute , where .

Explanation:

All we need to do is calculate the partial derivatives and add them together.

Example Question #3 : Divergence, Gradient, & Curl

Calculate the curl for the following vector field.

Explanation:

In order to calculate the curl, we need to recall the formula.

where , and  correspond to the components of a given vector field:

Now lets apply this to out situation.

Thus the curl is

Example Question #1 : Divergence, Gradient, & Curl

Compute , where .

Explanation:

All we need to do is calculate the partial derivatives and add them together.

Example Question #1 : Parameterization & Surface Integrals

Evaluate , where  is the region below the plane  , above the  plane and between the cylinders , and .

Explanation:

We need to figure out our boundaries for our integral.

We need to convert everything into cylindrical coordinates. Remeber we are above the  plane, this means we are above .

The region  is between two circles , and .

This means that

Example Question #1 : Parameterization & Surface Integrals

Evaluate , where  is the region below the plane  , above the  plane and between the cylinders , and .