Lesson 1

Question 1

What is vector integration?

Answer 1

The process of finding the integral of vector functions over a specified domain.

Question 2

True or False: The integral of a vector function results in a scalar quantity.

Answer 2

False

Question 3

What is the notation for integrating a vector function F?

Answer 3

∫ F(t) dt

Question 4

Fill in the blank: The integral of a vector field over a curve is called the _____ integral.

Answer 4

line

Question 5

What does the line integral of a vector field represent?

Answer 5

The work done by the field along a path.

Question 6

What is the formula for the line integral of a vector field F along a curve C?

Answer 6

∫_C F · dr

Question 7

True or False: The limits of integration for a line integral correspond to the endpoints of the curve.

Answer 7

True

Question 8

What is a vector function?

Answer 8

A function that assigns a vector to each point in its domain.

Question 9

What is the gradient of a scalar field?

Answer 9

A vector field that points in the direction of the greatest rate of increase of the scalar field.

Question 10

What is the divergence of a vector field?

Answer 10

A scalar that measures the rate at which ‘stuff’ is expanding or contracting at a point.

Question 11

Fill in the blank: The divergence of a vector field F is denoted as _____ F.

Answer 11

div

Question 12

What is the curl of a vector field?

Answer 12

A vector that represents the rotation of the field at a point.

Question 13

What is the notation for curl?

Answer 13

curl F or ∇ × F

Question 14

True or False: The curl of a vector field is a scalar quantity.

Answer 14

False

Question 15

What theorem relates the line integral of a vector field to the surface integral of its curl?

Answer 15

Stokes’ Theorem

Question 16

What does Green’s Theorem relate?

Answer 16

The line integral around a simple closed curve to a double integral over the plane region bounded by the curve.

Question 17

Fill in the blank: The divergence theorem relates surface integrals to _____ integrals.

Answer 17

volume

Question 18

What is the divergence theorem also known as?

Answer 18

Gauss’s Theorem

Question 19

What is the physical interpretation of the divergence of a vector field?

Answer 19

It represents the net flux out of an infinitesimal volume.

Question 20

What is the integral form of the divergence theorem?

Answer 20

∫∫_S F · dS = ∫∫∫_V div F dV

Question 21

Fill in the blank: In vector calculus, the term ‘flux’ refers to the _____ of a vector field through a surface.

Answer 21

flow

Question 22

What is the formula for the flux of a vector field F through a surface S?

Answer 22

Φ = ∫∫_S F · dS

Question 23

What is required to compute a line integral?

Answer 23

A parameterization of the curve and the vector field.

Question 24

True or False: The line integral depends on the path taken between two points.

Answer 24

True

Question 25

What is a conservative vector field?

Answer 25

A vector field where the line integral between two points is independent of the path taken.

Question 26

How can you determine if a vector field is conservative?

Answer 26

If its curl is zero everywhere in the domain.

Question 27

What is the potential function for a conservative vector field?

Answer 27

A scalar function whose gradient gives the vector field.

Question 28

Fill in the blank: The potential function is often denoted by _____ for a vector field F.

Answer 28

φ

Question 29

What does the notation ∇φ represent?

Answer 29

The gradient of the scalar potential function φ.

Question 30

What is the relationship between work done and line integrals in vector fields?

Answer 30

The work done is equal to the line integral of the force field along the path of motion.

Question 31

What is the significance of the path independence of conservative fields?

Answer 31

It allows for the definition of a potential energy associated with the field.

Question 32

True or False: All vector fields are conservative.

Answer 32

False

Question 33

What is the integral of the zero vector function?

Answer 33

The zero vector.

Question 34

What is the result of integrating a constant vector over a scalar variable?

Answer 34

The constant vector multiplied by the scalar variable plus a constant of integration.

Question 35

What is the physical interpretation of the line integral of a velocity vector field?

Answer 35

It represents the displacement of a particle moving along a path.

Question 36

What happens to the work done when moving against a conservative force?

Answer 36

The work done is positive.

Question 37

What is the geometric interpretation of the curl of a vector field?

Answer 37

It measures the tendency of the field to induce rotation around a point.

Question 38

Fill in the blank: In physics, the concept of work done by a force vector field is calculated using _____ integrals.

Answer 38

line

Question 39

What does the term 'vector field' refer to in mathematics?

Answer 39

A function that assigns a vector to every point in a subset of space.

Question 40

True or False: The integral of a vector function can be evaluated using standard techniques for scalar functions.

Answer 40

True

Question 41

What is the primary application of vector integration in physics?

Answer 41

To calculate work done by forces and flow of fluids.