###### Learning Outcomes:

On completion of this module students should be able to

1. Write down parametric equations for lines and planes, and perform standard calculations based

on these equations (e.g. points/lines of intersection, condition for lines to be skew);

2. Compute the Frenet-Serret vectors for an arbitrary differentiable curve;

3. Compute the series expansion of important functions using Taylor's theorem;

4. Differentiate scalar and vector fields expressed in a Cartesian framework;

5. Perform operations involving div, grad, and curl;

6. Perform line, surface, and volume integrals. The geometric objects involved in the integrals

may be lines, arbitrary curves, simple surfaces, and simple volumes, e.g. cubes, spheres,

cylinders, and pyramids;

7. State precisely and prove Gauss's and Stokes's theorems;

8. Derive corollaries of these theorems, including Green's theorems and the necessary and sufficient condition for a vector fueld to be derivable from a potential;

9. Compute the scale factors for arbitrary orthogonal curvilinear coordinate systems;

10. Apply the formulas for div, grad, and curl in arbitrary orthogonal curvilinear coordinate systems;