1. Sketch and describe regions in R^2 and R^3 in Cartesian, polar, cylindrical and spherical coordinates as applicable
2. Calculate integrals of multivariable functions
- determine whether a given vector field is conservative and if so identify its potential
- parametrise curves and compute line integrals, including in the applications of work and mass
- compute double, surface and triple integrals, including in the applications of mass, area, volume and flux
3. apply the theorems of Gauss (Divergence), Green and Stokes
- compute the divergence and curl of a vector field and explain their physical meanings
- decide which type of integral is relevant for each application
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Vector Calculus extends the single variable calculus of Calculus 1 and 2 into multivariable and vector calculus. The course deals extensively with integration; we shall cover double, triple, surface and line integrals. In the contexts of double and triple integrals we shall carry out coordinate changes, While our major focus is on shifts between Cartesian and polar/cylindrical/spherical coordinates, we shall also look at special one-off cases of coordinate changes. A key feature of this section of the course is the Jacobian. Applications we shall look at include area (double, surface), mass (line, double, surface, triple), work done (line) and flux (surface). We shall look at vector fields and line integrals of vector fields where path independence is an important concept. Using the concepts of div and curl, we shall close the course off with the powerful vector calculus theorems: Green’s, Stokes’ and Divergence, which extend the Fundamental Theorem of Calculus into higher dimensional spaces.
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