**Integral Calculus**

1.3.5 Applied to Curls

1.3.4 Applied to Divergences

1.3.3 Applied to Gradients

1.3.2 Fundamental Theorem of Calculus

1.3.1 Types of integrals

**Types of integrals**

Line

Surface

Volume

open

closed

dl depends on the path you take.

closed paths start and stop at the same location

da depend on the surface

closed surfaces have distinct outsides and insides, think balloon or box

Can 'act' on a vector or scalar

Fundamental theorem of Calculus

This says:

The integral of a derivative of a function, over a specific interval. is given by the value of that function at the boundary conditions.

So, we need to talk about 'derivative' and vector functions, and the idea of that interval/boundary

**Gradients**

You will note that the fundamental theorem of calculus was just one direction.

An easy upgrade is to make it a gradient, that takes care of our

first derivative option

.

recall that it doesn't matter the path you take, but that you evaluate at the boundaries,

Also the idea of intervals and boundaries.

first option, is a

line, with end points

.

so these two link together and give

it only makes sense to talk about how things change along a path, ie what their gradient is.

**Divergence**

Our first option was

path with endpoints

boundaries,

let us skip

surface with line

boundaries (Stoke's theorem)

and go to a

volume with surface

boundaries and derivative a

divergence

This is Green's theorem

(or Gauss's / divergence theorem)

The idea you can count up what is inside a volume by doing a volume integral, or by counting what goes and comes through its surface boundary. (things that change both of those are describe by divergence)

**Curl**

back to

surface with line

boundaries, and derivative being a

curl

Stoke's Theorem

Here we can measure how well things curl on a surface and that should equal how much things change on the edges of that surface.

Note it doesn't depend on what the surface is just the boundaries of that surface

and if you close the surface it is zero.

Integrating the vector v along path 1 or 2 results in

A same value

B different value

C v should be a scalar to

do this.

D No idea, please

review

Which is the right dl listed below

A dl is dz on path 1

B dl is dy on path 1

C dl is both dz and dy

simultaneously on path 1

D dl is both dz and dy

simultaneously on path 2

If the final location of both these paths is (5,4), which is the correct gradient integral for T along the corresponding

complete

path?

D

C

B

A

**Practice**

Show that the gradient theorem is true on both paths to the location (1,1,1), path 2 is a straight line from the origin to the final point.

In calculating this volume integral, which is not true?

A

D

C

B

What is the volume integral of

over this volume shown here?

Prove the divergence theorem for this function over the volume of a cube side 1, at the origin on the axis

3/2

consider the unit cube in the primary quadrant, and take the side that is at (1,y,z)

What will the integral of v*da look like for this vector

D

C

B

A

consider the unit cube in the primary quadrant, and take the side that is at (x,y,0)

What will the integral of v*da look like for this vector

D

C

B

A

E

none of these

E

none of these

Lets finish this example by proving Stokes's theorem for this same function on only the side (x,1,z) using direction giving out as positive.

5/3

"I didn't really understand the math behind the fundamental theorem of Divergences"

"Could you go more into depth for the Corollaries of the Fundamental Theorem of Gradients?"

"Why does equation 1.55 work?"

"I don't understand what it means for the geometry to become closed."