Multivariate calculus Lecture note 5 Outline 1. Multivariate functions in Euclidean space 2. Continuity 3. Multivariate differentiation 4. Differentiability 5. Higher order derivatives 6. Implicit functions 1
Functions between Euclidean spaces Recall that a function : is a rule that assigns for every element in a unique element in. Key terms: domain, range, image The most common function that you will come across in economics is :. Some examples:, (linear production function), (Cobb-Douglas utility) The usual extension is to enlarge the dimension of the range : For example, if is a vector of a consumer s demands (i.e., each dimension is the demand for a different good), then we can have where is the vector of prices. Geometric reprazentation You can forget about visualising virtually all such functions. However there is an interesting class where it s worth making the effort. These are functions of the form :. We can plot contours in 2
Example:, Example:, 3
Linear functions are an important class. A function : is linear if and only if: For all scalars, and all, All linear functions can be expressed in the form where is a matrix. Continuity Intuitive diagrams of what does and does not constitute continutity 4
Formal definition: is continuous at a point if and only if: Diagrammatic elucidation Example: prove that 2 is continuous at 3. Example: prove that the function 1 2 2 2 is not continuous at the point 2. 5
A function is continuous if it is continuous at all points in its domain. NB: if the range is multidimensional, then must be continuous in each dimension is equivalent Example: prove that is continuous Theorem: let and be continuous functions. Then, and are also continuous functions. Examples 6
Other properties of functions A function is surjective if an element of the domain that maps into every element of the range Examples A function is injective if. This means that it is invertible. Examples A bijective function is both. 7
Calculus of several variables Let :. Definition of a partial derivative at a point : NB: if it exists! Graphical interpretation,, lim,,,,,, All the standard rules of differentiation apply. The only difference is that you treat all other variables as constants. Examples:,, 3 8
Partial derivatives as marginal products Elasticities in derivative form Differentiability Distinction between a left limit and a right limit Theorem: a function is differentiable if and only if it is continuous and every left limit is equal to every right limit Graphical examples of violations of differentiability 9
Differentiation as a linear approximation Estimating using,, Geometric logic Examples 10
The total differential Often in economics, you will see the following trick Note the different meaning of and Can also be written as (explain notation) How does this relate to the gradient of contours? Application to consumer theory 11
The multivariate chain rule Suppose that (all vectors) and. Then: Example:, 2, 2,. Find 12
Higher order derivatives Higher order derivatives are defined in the same way as univariate calculus. Example: 2. Find Definition of a Hessian Example: evaluate the Hessian of the above example Young s theorem: 13
Implicit functions On lots of occasions, we have an expression of the form, 0 where we cannot isolate Example: 3 0 However we still want to evaluate Again, the methods are entirely analogous to the univariate case. Do the example above This will come in extremely handy when we do comparative statics in the optimisation section, which I m sure you are awfully excited to see. 14
Summary The most important things to take from this lecture note are: How to specify multivariate functions Continuity and differentiability in multivariate functions How to do partial differentiation in explicit and implicit functions These tools will underlie optimisation, which is the most important tool for an economist. 15