Differential Calculus in Several Variables

Week 1: Functions of several variables and examples; continuity; concept of distances in higher dimension; examples; how to define differentiable functions.
Week 2: Directional derivatives as direct generalization from one variable; drawbacks; definition of differentiable functions of several variable; examples; matrix of a linear transformation; Jacobian matrix; recovering directional derivative from derivative; chain rule.
Week 3: Comparison with one variable calculus: version of MVT, higher derivatives and Taylor’s formula; Sufficient condition for equality of mixed derivatives; extremas for real valued function and special case for functions with two variables.
Week 4: Statement of Implicit Function Theorem and Inverse Function Theorem.; deriving Inverse Function Theorem form Implicit Function Theorem; Illustration of Implicit Function Theorem for function of two variables; examples from solutions of ordinary differential equations; proof of Implicit Function Theorem; Lagrange’s Multipliers.