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1.3.6 - 1.4.2 & 1.6

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Richard Datwyler

on 25 September 2018

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Transcript of 1.3.6 - 1.4.2 & 1.6

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"
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