# Calculus 3 : Curl

## Example Questions

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### Example Question #1 : 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 : 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 #78 : Line Integrals

Find the curl of the force field

Explanation:

Curl is probably best remembered by the determinant formula

, which is used here as follows.

### Example Question #4 : Curl

Let  be any arbitrary real valued vector field. Find the

Explanation:

Take any field, the curl gives us the amount of rotation in the vector field. The purpose of the divergence is to tell us how much the vectors move in a linear motion.

When vectors are moving in circular motion only, there are no possible linear motion. Thus the divergence of the curl of any arbitrary vector field is zero.

### Example Question #5 : Curl

Evaluate the curl of the force field .

Explanation:

To evaluate the curl of a force field, we use Curl

. Start

Evaluate along the first row using cofactor expansion.

. Evaluate partial derivatives. All terms except the 2nd to last one are .

### Example Question #3 : Curl

Calculate the curl  of the following vector:

Explanation:

The curl of a vector

is defined by the determinant of the following 3x3 matrix:

For the given vector, we can calculate this determinant

### Example Question #4 : Curl

Given that F is a vector function and is a scalar function, which of the following operations results in a scalar?

Explanation:

For each of the given expressions:

- The divergence of a scalar function does not exist, so this expression is undefined.

- The dot product of a vector function is a scalar, so the gradient of the term in parenthesis results in a vector.

- The divergence of a vector function is a scalar.  Taking the divergence of the term in parenthesis would be taking the divergence of a scalar, which doesn't exist.  This expression is undefined.

- The gradient of a scalar function is a vector.  Thus, the curl of the term in parenthesis is also a vector.

- The term in parenthesis is the curl of a vector function, which is also a vector.  Taking the divergence of the term in parenthesis, we get the divergence of a vector, which is a scalar.

### Example Question #5 : Curl

Given that is a vector function and is a scalar function, which of the following expressions is undefined?

Explanation:

The cross product of a scalar function is undefined.  The expression in the parenthesis of:

is the cross product of a scalar function, therefore the entire expression is undefined.

For the other solutions:

- The cross product of a vector is also a vector, and the divergence of a vector is defined. This expression is a scalar.

- The gradient of a scalar is a vector, and the divergence of a vector is defined.  This expression is also a scalar.

- The divergence of a vector is scalar, and the gradient of a scalar is defined.  This expression is a vector.

- The gradient of a scalar is a vector, and the curl of a vector is defined.  This expression is a vector.

### Example Question #6 : Curl

Compute the curl of the following vector function:

Explanation:

For a vector function , the curl is given by:

For this function, we calculate the curl as:

### Example Question #7 : Curl

Compute the curl of the following vector function: