Multivariable Calculus Chapter 2.3

Front 1

Partial Derivatives

Back 1

Let U⊂Rⁿ be an open set and suppose f:U⊂Rⁿ→R is a real valued function. Then δf/δx₁,...,δf/δxn the partial derivatives of f with respect to the first, second, ..., nth variable, are the real valued function of n variables which at the point (x₁,...,xn)= x, are defined by

δf/δxj = lim [f(x+h) -f(x)]/h

h→0

Front 2

Tangent plane

Back 2

z= f(x₀,y₀) + [δf/δx (x₀,y₀)] (x-x₀) + [δf/δy (x₀,y₀)] (y-y₀)

This is the tangent plane of the graph of f at the point (x₀,y₀)

Front 3

Differentiable: Two Variables

Back 3

Let f:R²→R. We say f is differentiable at (x₀,y₀), if δf/δx and δf/δy exist at (x₀,y₀) and if

{f(x,y) - f(x₀,y₀) - [δf/δx (x₀,y₀)] (x-x₀) + [δf/δy (x₀,y₀)] (y-y₀)} /‖(x,y)-(x₀,y₀)‖ →0

as (x,y) →(x₀,y₀)

Front 4

Differentiable, n Variables, m Functions

Back 4

We say that f is differentiable at x₀∈U if the partial derivatives of f exist at x₀ and if

limit ‖f(x) - f(x₀) - T(x-x₀)‖/‖x-x₀‖ =0 as x→x₀

T=Df(x) is a m×n matrix with elements δfi/δxj evaluated at x₀ and T(x-x₀) means the product of T with x-x₀ (regarded as a column matrix)

Front 5

Gradient

Back 5

Consider the special case f:U⊂Rⁿ→R. Here Df(x) is a 1×n matrix:

Df(x) = [δf/δx₁ , ... ,δf/δxn]

We can form the corresponding vector (δf/δx₁ , ... ,δf/δxn), called the gradient of f and denoted by ∀f or grad f