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0/1. Revision and
mathematical Preliminaries
0.1 Previous “Electricity & Magnetism”
1.1 Scalars & vectors
1.2.1 GRAD, 1.2.2 DIV, 1.2.3 CURL
1.2.4 Vector calculus
1.2.5 Integral theorems
1.2.6 Co-ordinate systems
1.3 Dirac delta
1.4 Laplace’s and Poisson’s equations
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Revision
- DIV, GRAD & CURL
- Laplacian
- Vector identities
- Divergence & Stokes’ theorems
(*) There is a “formula sheet” in the first set of notes.You are expected to know the formulae in Cartesiancoordinates, but will not be expected to remember orprove those for cylindrical and spherical polarcoordinates.
Grad
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Divergence
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+ ve -ve
Eg v = xi + yj Eg v = -xi - yj
Curl
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CURL is out of the page
Eg v = -yi + xj
Vector identities
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Divergence theorem
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Stokes theorem
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Normal into
the page
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Old and new…
- Polar coordinates
- Dirac delta “function”
- Laplace’s equation & Poisson’s equation
- Uniqueness theorem for Poisson’s equation
- General solution to Poisson’s equation
3D curvilinear coordinates
We will often wish to use 3D curvilinear
coordinate systems such as cylindrical polar
coordinate.
The most important point is that the formulae
for GRAD, DIV, CURL and the LAPLACIAN are
modified because the basis vectors depend
on position.
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Cylindrical & Spherical Polars
Cylindrical coordinates Spherical coordinates (MATHS!!)
Divergence - cylindrical polars
- you will not be expected to remember these
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Dirac delta
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This, the Dirac delta, provides our route to
mathematically defining the density of an
infinitely small object with has finite
charge (or current).
Gaussian function
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Point particles
Eg. Charge density of a set of point
particles with charge qi and positions ri
NB. delta has dimensions, (length)-n
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Important identities…
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Old and new…
- Polar coordinates
- Dirac delta “function”
- Laplace’s equation & Poisson’s equation
- Uniqueness theorem for Poisson’s equation
- General solution to Poisson’s equation
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Blended Learning…
- (Previous courses and notes)
- Supporting videos
- The notes and recommended texts
- Critically, the example sheets (for your tutorial) and the online exercises
- The tutorial
- Synchronous session and discussions
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BL notes & books;
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BL notes & books;
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BL Blackboard examples;
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BL Tutorial examples;
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BL synchronous session;
Before your tutorial we will meet and in
that (weekly) session, address common
queries that have been raised and review
the examples..
