Math Methods for Scientist 1b

Learning Outcomes

Learning Outcomes

Having successfully completed this module you will be able to:

• Apply the divergence theorem and Stokes' theorem
• Evaluate partial derivatives and find critical points of functions of two variables
• Evaluate line integrals and fluxes of vector fields over curves and surfaces
• Express curves and surfaces in both parametric and implicit form
• Evaluate integrals of simple functions over simple regions of the plane and simple volumes
• Solve second order linear equations with constant coefficients
• Evaluate the gradient of a scalar field and the divergence and curl of a vector field
• Identify and solve first order ODEs that are separable, linear or exact

Functions of two or more variables:

Evaluate partial derivatives, find critical points, and, for functions of two variables, classify them.

Multiple Integrals of a scalar function in (2 and 3 dimensions):

Evaluate integrals of simple functions over regions in plane bounded by graphs of simple functions, either directly or by change of coordinate system. Evaluate integrals over volumes bounded by planes, spheres and cylinders, using cylindrical and polar coordinates.

Vector Calculus:

Gradients, divergences and curls.

Curves and line integrals:

Express, in simple cases, curves given parametrically. Evaluate lengths of curves in 2 and 3 dimensions. Evaluate integrals of scalar functions along curves with respect to arc-length. Evaluate

the integral of the tangential component of a vector field along a curve. Conservative fields.

Surfaces:

Integration of normal components of a vector field or of a scalar field over surfaces described parametrically.

The divergence theorem and and Stokes' theorem and their application.

Differential equations

Types of ordinary differential equation. Solving simple differential equations, separation of variables, integrating factors and first order linear ordinary differential equations. Exact differential equations. Second order differential equations. Homogeneous linear ordinary differential equations with constant coefficients. Free and forced damped harmonic oscillator.

Lectures, small group tutorials, private study. The method of delivery in lectures will be “chalk and talk”, using skelatal lecture notes.

Summative

This is how we’ll formally assess what you have learned in this module.

Referral

This is how we’ll assess you if you don’t meet the criteria to pass this module.