## Mathematical Methods & Applications

(sem 1 of 33098)

**• Some revision of partial differentiation**

See Tutorial 11, and Tutorial 11 Supplement on partial differentiation (from 1st year Maths). If you need revision of this topic, it is probably best to look at the supplementary tutorial first. Here is a pdf copy of Tutorial 11 and a link to the ‘clickable version’ of the Tutorial 11 Supplementary (in which you can click on the green bits).

**• Past January Continuous Assessment Test – Questions & Solutions**

**Sample Exam Questions Handout**

Higher resolution copies: **Sample Exam 1** | **Sample Exam 2** | **Sample Exam 3**

**• Presentation Slides**

h1 | h2 | h3 | h4 | h5 | h6 | h7 | h8 | h9 | h10

**• Presentation Summaries**

h1 | h2 | h3 | h4 | h5 | h6 | h7 | h8 | h9 | h10

## Vector Calculus

• Review of fundamental concepts. Scalar, vector and conservative fields. Grad, divergence, flux and curl.

Handouts 1, 2 & 3

Tutorial A scan (with extra solutions) | Tutorial A (interactive copy) (grad & direction derivatives)

Tutorial B scan (with extra solutions) | Tutorial B (interactive copy) (div & curl introduction)

Tutorial C (interactive copy) (using the Laplacian operator)

Note on the Laplacian of Scalar and Vector Fields

• The divergence theorem and Stoke’s theorem. The Laplacian and curvilinear coordinates. Examples from electrostatics, magnetism, fluid dynamics, mechanics, heat flow.

## Determinants and Matrices

• Basic definitions and operations.

Tutorial D scan | Tutorial D (interactive copy) (matrix multiplication)

• Cramer’s rule and Laplace expansion. Rank, linear independence, elementary row operations and matrices in echelon form. Properties of determinants. Special matrices and matrix inversion. Eigenvalues and eigenvectors.

• Applications in electrical circuits, rotation of co-ordinates, transmission through single and cascaded linear systems (such as in optics and electronics)

Handouts 7, 8 & 9

**Main Tutorial 3**

## Differential Equations

• Review of ordinary differential equations. Important partial differential equations (PDE’s). Solution of PDEs and the role of arbitrary functions. Separation of variables. Examples drawn from a broad range of physics