Line Integrals over Vector Fields

Question 1

Define line integral.

Answer 1

A line integral calculates the integral of a function along a curve.

Question 2

True or false: Line integrals can be used for scalar and vector fields.

Answer 2

TRUE

Question 3

What does a conservative vector field imply?

Answer 3

A conservative vector field has a potential function and is path-independent.

Question 4

Fill in the blank: A vector field is conservative if its curl is ______.

Answer 4

Zero

Question 5

Define potential function.

Answer 5

A potential function is a scalar function whose gradient gives a vector field.

Question 6

True or false: Every conservative vector field has a potential function.

Answer 6

TRUE

Question 7

What is the relationship between line integrals and conservative fields?

Answer 7

In conservative fields, line integrals depend only on endpoints, not the path.

Question 8

Fill in the blank: The work done by a conservative force is equal to the change in ______.

Answer 8

Potential energy

Question 9

Define path independence in the context of line integrals.

This applies to Conservative vector fields

Answer 9

Path independence means the integral value is the same regardless of the path taken.

Question 10

What is the gradient of a potential function?

Answer 10

The gradient of a potential function gives the vector field associated with it.

Question 11

True or false: A non-conservative vector field has a potential function.

Answer 11

FALSE

Question 12

Fill in the blank: The integral of a conservative vector field over a closed path is ______.

Answer 12

Zero

Question 13

What is the work-energy theorem in conservative fields?

Answer 13

The work done by conservative forces equals the change in kinetic energy.

Question 14

Define curl in relation to vector fields.

Answer 14

Curl measures the rotation of a vector field at a point.

Question 15

Fill in the blank: If a vector field is irrotational, its curl is ______.

Answer 15

Zero