Calculus 3 : Curl

Study concepts, example questions & explanations for Calculus 3

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Example Questions

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Example Question #1 : Curl

Calculate the curl for the following vector field.

Possible Answers:

Correct answer:

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.

Possible Answers:

Correct answer:

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 #3 : Curl

Find the curl of the force field

Possible Answers:

All of the other answers

None of the other answers

Correct answer:

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 

Possible Answers:

Correct answer:

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 #4 : Curl

Evaluate the curl of the force field .

Possible Answers:

Correct answer:

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 #5 : Curl

Calculate the curl  of the following vector:

Possible Answers:

Correct answer:

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 #7 : Curl

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

Possible Answers:

Correct answer:

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 remaining answer is:

 - 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 #6 : Curl

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

Possible Answers:

Correct answer:

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 #7 : Curl

Compute the curl of the following vector function:

Possible Answers:

Correct answer:

Explanation:

For a vector function , the curl is given by:

For this function, we calculate the curl as:

Example Question #10 : Curl

Compute the curl of the following vector function:

Possible Answers:

Correct answer:

Explanation:

For a vector function , the curl is given by:

For this function, we calculate the curl as:

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