**1.3.6 - 1.4.2 & 1.6**

1.3.6 Integration by parts

1.4.1 Spherical Polar Coordinates

1.4.2 Cylindrical Coordinates

1.6.1 Helmholtz Theorem

1.6.2 Potentials

Integration by parts

a bit better would be

moving the derivative around turns out to be a nice tool to have, so recall this later...

**Spherical Polar Coordinates**

many physical situations have symmetries, where ever a good symmetry exists it is nice to have a coordinate system to match.

We will transform our cartesian (x,y,z) coordinates into a spherical polar coordinate (radius, polar, azimuthal)

The question is: how do they relate.

**Definitions**

Definitions

Unit vectors

Infinitesimals.

Derivatives

0

o

r

z

y

x

**Unit vectors**

0

o

r

z

y

x

o

0

r

Infinitesimal elements

r

dr

r

d0

r d0

0

d0

r sin 0 do

Area

surface of a sphere?

surface in the x y plane?

surface in the x z plane?

surface in the y z plane?

**Differentials**

Gradient:

Divergence:

Curl:

Laplacain:

front of your text. go look them up.

Cylindrical Coordinates

Definitions

back of book

Unit vectors

back of book

**Infinitesimals**

Front of book

**Differentials**

Front of book

What is the gradient of

in the polar direction?

**D**

**C**

**B**

**A**

What is the gradient of ?

Test the gradient theorem for this function following this path.

1

1

1

What is the divergence of

D

C

B

A

Prove the divergence theorem over the area shown here.

1

1

1

-1

1

**Helmholtz theorem**

Does knowing the divergence and curl of a vector uniquely determine the vector?

No, need boundary conditions.

**Potentials**

We can use rewrite a vector field in terms of a gradient of a scalar and a curl of a vector

we use this because we can use gauge theorems later and define vector fields in terms of potentials

"Could you just review taking a surface integral"

"Can you cover the integration by parts quickly"

"Could we go over the math behind the partial derivatives involving the unit vectors in spherical coordinates and what it means physically?"

"transformation matrix Cartesian to cylindrical and Cartesian to spherical"