Local Linear Approximation for Functions of Several Variables

Functions of One Variable When we zoom in on a “sufficiently nice” function of one variable, we see a straight line.

Functions of two Variables

When we zoom in on a “sufficiently nice” function of two variables, we see a plane.

(a,b)(a,b) Describing the Tangent Plane To describe a tangent line we need a single number--- the slope. What information do we need to describe this plane? Besides the point (a,b), we need two numbers: the partials of f in the x- and y- directions. Equation?

Describing the Tangent Plane We can also write this equation in vector form. Write x = (x,y), p = (a,b), and Dot product! Gradient Vector!

General Linear Approximations Why don’t we just subsume F(p) into L p ? Linear--- in the linear algebraic sense. In the expression we can think of As a scalar function on  2 :. This function is linear.

General Linear Approximations In the expression we can think of As a scalar function on  2 :. This function is linear. Note that the expression L p (x-p) is not a product. It is the function L p acting on the vector (x-p).

Linear We must understand Linear Vector Fields: Linear Transformations from Differentiability Vector Fields To understand Differentiability in Vector Fields

Linear Functions A function L is said to be linear provided that Note that L(0) = 0, since L(x) = L (x+0) = L(x)+L(0). For a function L:  m →  n, these requirements are very prescriptive.

Linear Functions It is not very difficult to show that if L:  m →  n is linear, then L is of the form: where the a ij ’s are real numbers for j = 1, 2,... m and i =1, 2,..., n.

Linear Functions Or to write this another way... In other words, every linear function L acts just like left- multiplication by a matrix. Though they are different, we cheerfully confuse the function L with the matrix A that represents it! (We feel free to use the same notation to denote them both except where it is important to distinguish between the function and the matrix.)

One more idea... Suppose that A = (A 1, A 2,..., A n ). Then for 1  j  n What is the partial of A j with respect to x i ? A j (x) = a j1 x 1 +a j 2 x a j n x n

The partials of the A j ’s are the entries in the matrix that represents A! One more idea... Suppose that A = (A 1, A 2,..., A n ). Then for 1  j  n A j (x) = a j1 x 1 +a j 2 x a j n x n

Local Linear Approximation

What can we say about the relationship between the matrix DF(p) and the coordinate functions F 1, F 2, F 3,..., F n ? Quite a lot, actually... Suppose that F:  m →  n is given by coordinate functions F=(F 1, F 2,..., F n ) and all the partial derivatives of F exist at p in  m and are continuous at p then... affine there is a matrix L p such that F can be approximated locally near p by the affine function L p will be denoted by DF(p) and will be called the Derivative of F at p or the Jacobian matrix of F at p.

A Deep Idea to take on Faith I ask you to believe that for all i and j with 1  i  n and 1  j  m This should not be too hard. Why? Think about tangent lines, think about tangent planes. Considering now the matrix formulation, what is the partial of L j with respect to x i ?

The Derivative of F at p ( sometimes called the Jacobian Matrix of F at p ) Notice the two common nomeclatures for the derivative of a vector – valued function.

In Summary... For all x, we have F(x)=DF(p) (x-p)+F(p)+E(x) where E(x) is the error committed by DF(p) +F(p) in approximating F(x). If F is a “reasonably well behaved” vector field around the point p, we can form the Jacobian Matrix DF(p). How should we think about this function E(x)?

E(x) for One-Variable Functions But E(x)→0 is not enough, even for functions of one variable! What happens to E( x ) as x approaches p ? E(x) measures the vertical distance between f (x) and L p (x)

Definition of the Derivative A vector-valued function F=(F 1, F 2,..., F n ) is differentiable at p in  m provided that 1)All partial derivatives of the component functions F 1, F 2,..., F n of F exist on an open set containing p, and 2)The error E(x) committed by DF(p)(x-p)+F(p) in approximating F(x) satisfies (Book writes this as.)