Stoke's Theorem

Question 1

Define surface integral.

Answer 1

An integral that computes the total of a function over a surface in three-dimensional space.

Question 2

What does the unit normal vector represent?

Answer 2

A vector perpendicular to a surface with a magnitude of one, indicating direction.

Question 3

True or false: The unit normal vector can vary across a surface.

Answer 3

TRUE

The unit normal vector changes depending on the surface’s curvature.

Question 4

Fill in the blank: The Stokes’ theorem relates surface integrals to _______.

Answer 4

line integrals

Question 5

What is the formula for Stokes’ theorem?

Answer 5

The integral of a vector field over a surface equals the integral over its boundary curve.

Question 6

Define curl in vector calculus.

Answer 6

A measure of the rotation of a vector field at a point.

Question 7

True or false: Stokes’ theorem applies only to closed surfaces.

Answer 7

FALSE

Stokes’ theorem applies to surfaces bounded by a simple, closed curve.

Question 8

What is the relationship between curl and line integrals?

Answer 8

The line integral of a vector field around a curve equals the surface integral of its curl.

Question 9

Fill in the blank: The flux through a surface is calculated using a _______.

Answer 9

surface integral

Question 10

Define divergence.

Answer 10

A scalar measure of the rate at which a vector field spreads out from a point.

Question 11

What does the normal vector indicate?

Answer 11

The orientation of a surface at a given point, crucial for surface integrals.

Question 12

True or false: The surface integral of a constant function is simply the area of the surface.

Answer 12

TRUE

This holds when integrating a constant over a flat surface.

Question 13

What is the boundary of a surface in Stokes’ theorem?

Answer 13

The curve that encloses the surface, where the line integral is evaluated.

Question 14

Fill in the blank: The flux of a vector field through a surface depends on its _______.

Answer 14

orientation

Question 15

Define Green’s theorem.

Answer 15

A special case of Stokes’ theorem in the plane, relating line integrals and area integrals.

Question 16

What is the significance of the normal vector in surface integrals?

Answer 16

It determines the direction of the surface area element in the integral.

Question 17

True or false: Stokes’ theorem can be applied to non-smooth surfaces.

Answer 17

FALSE

The theorem requires surfaces to be piecewise smooth.

Question 18

What is a parameterization of a surface?

Answer 18

A mathematical representation of a surface using parameters to describe its points.

Question 19

Fill in the blank: The surface integral of a vector field is denoted as _______.

Answer 19

∬_S F · dS