Mathematical Analysis 2: Key skills

WARNING

The information in this section is being updated.

Sequences and series of functions

• Distinguish point-wise and uniform convergence
• Calculate Taylor series of functions
• Calculate radius of convergence of a power series
• Differentiate / integrate a power series

Functions in ${\mathbb{R}}^n$

• Calculate partial derivatives of a function of several variables
• Calculate the gradient of a scalar field ($\mathrm{\nabla }f$)
• Calculate the directional derivatives
• Calculate the Jacobian matrix of a general function
• Use the chain rule for these higher dimensional derivatives
• Calculate curl of a vector field ($\mathrm{\nabla }\times F$)
• Calculate divergence of a vector field ($\mathrm{\nabla }\cdot F$)
• Visualize and sketch vector fields
• Visualize and sketch level sets
• Find the normal to a level set surface
• Identify connected and simply connected regions

Extrema

• Find stationary points of a scalar field
• Use the Hessian matrix to classify stationary points
• Apply the Lagrange multiplier method to find the extrema of a constrained problem
• Determine the second-order Taylor formula for a scalar field

Multiple integrals

• Evaluate 2D/3D multiple integrals by repeated integration
• Manipulate the description of 2D/3D regions in ways that are useful for integrating
• Exchange the order of integration
• Perform change of variable for multiple integrals
• Use polar coordinates, cylindrical coordinates, spherical coordinates

Curves / paths

• Work with parametric curves
• Identify and construct parametric curves
• Evaluate the length of a path
• Evaluate the path integral of a scalar field
• Evaluate the path integral of a vector field

Surfaces

• Work with parametric surfaces
• Identify and construct standard parametric surfaces
• Calculate the fundamental vector product of a parametric surface
• Evaluate the area of a surface
• Evaluate the surface integral of a scalar field
• Evaluate the surface integral of a vector field (flux integral)

Applications, etc.

• Calculate the centre of mass / centroid
• Determine if a vector field is conservative
• Construct a potential for a conservative vector field
• Apply the 1st and 2nd fundamental theorems of calculus for path integrals
• Apply Green's theorem
• Apply Stokes' theorem
• Apply Gauss' theorem