Mathematics  >  Mathematics for IIT JAM, CSIR NET, UGC NET  >  Derivative as a Linear Transformation - Differential Calculus, CSIR-NET Mathematical Sciences

Derivative as a Linear Transformation - Differential Calculus, CSIR-NET Mathematical Sciences - Notes | Study Mathematics for IIT JAM, CSIR NET, UGC NET - Mathematics

1 Crore+ students have signed up on EduRev. Have you?

Redefining the derivative

Matrices appear in many different contexts in mathematics, not just when we need to solve a system of linear equations. An important instance is linear approximation. Recall from your calculus course that a differentiable function f can be expanded about any point a in its domain using Taylor's theorem. We can write

Derivative as a Linear Transformation - Differential Calculus, CSIR-NET Mathematical Sciences - Notes | Study Mathematics for IIT JAM, CSIR NET, UGC NET - Mathematics

where c is some point between x and a. The remainder term Derivative as a Linear Transformation - Differential Calculus, CSIR-NET Mathematical Sciences - Notes | Study Mathematics for IIT JAM, CSIR NET, UGC NET - Mathematics is the "error"made by using the linear approximation to f at x = a,

Derivative as a Linear Transformation - Differential Calculus, CSIR-NET Mathematical Sciences - Notes | Study Mathematics for IIT JAM, CSIR NET, UGC NET - Mathematics

That is, f (x) minus the approximation is exactly equal to the error (remainder) term. In fact, we can write Taylor's theorem in the more suggestive form

f (x) = f (a) + f '(a)(x a) + ∈(x, a),

where the remainder term has now been renamed the error term (x; a) and has the important property

Derivative as a Linear Transformation - Differential Calculus, CSIR-NET Mathematical Sciences - Notes | Study Mathematics for IIT JAM, CSIR NET, UGC NET - Mathematics

(The existence of this limit is another way of saying that the error term \looks like" (x - a)2.) This observation gives us an alternative (and in fact, much better) de nition of the derivative:

De nition: The real-valued function f is said to be differentiable at x = a if there exists a number A and a function ∈(x, a) such that

f (x) = f (a) + A(x - a) + ∈(x, a),

where

Derivative as a Linear Transformation - Differential Calculus, CSIR-NET Mathematical Sciences - Notes | Study Mathematics for IIT JAM, CSIR NET, UGC NET - Mathematics

Remark: the error term Derivative as a Linear Transformation - Differential Calculus, CSIR-NET Mathematical Sciences - Notes | Study Mathematics for IIT JAM, CSIR NET, UGC NET - Mathematics clearly depends on a, and it depends on x as well since the number c varies with x.

Theorem: This is equivalent to the usual calculus definition.

Proof: If the new de nition holds, then if we compute f '(x) by the usual de nition, we find

Derivative as a Linear Transformation - Differential Calculus, CSIR-NET Mathematical Sciences - Notes | Study Mathematics for IIT JAM, CSIR NET, UGC NET - Mathematics

and A = f '(a) according to the standard de nition. Conversely, if the standard de nition of differentiability holds, then we can de ne ∈(x, a) to be the error made in the linear
approximation:

∈(x, a) = f (x) - f (a) - f '(a)(x - a)

Then

Derivative as a Linear Transformation - Differential Calculus, CSIR-NET Mathematical Sciences - Notes | Study Mathematics for IIT JAM, CSIR NET, UGC NET - Mathematics

so f can be written in the new form, with A = f '(a).

Example: Let f (x) = 4 + 2x - x2, and let a = 2. So f (a) = f (2) = 4; and f '(a) = f '(2) = 2 - 2a = - 2.. Now subtract f (2) + f '(2)(x - 2) from f (x) to get

4 + 2x - x2 (4 - 2(x - 2)) = -4 + 4x - x2 = (x - 2)2.

This is the error term, which is quadratic in x - 2, as advertised. So 8 - 2x(= f (2) + f '(2)(x - 2)) is the correct linear approximation to f at x = 2.

Suppose we try some other linear approximation - for example, we could try f (2) - 4(x - 2) = 12 - 4x: Subtracting this from f (x) gives -8 + 6x - x2 = -2(x - 2) - (x - 2)2, which is our new error term. But this won't work, since

Derivative as a Linear Transformation - Differential Calculus, CSIR-NET Mathematical Sciences - Notes | Study Mathematics for IIT JAM, CSIR NET, UGC NET - Mathematics

which is clearly not 0. The only \linear approximation" that leaves a quadratic remainder as the error term is the one formed in the usual way, using the derivative.

Exercise: Interpret this geometrically in terms of the slope of various lines passing through the point (2, f (2)).

Generalization to higher dimensions

Our new de nition of derivative is the one which generalizes to higher dimensions. We start with an

Example: Consider a function from R2 to R2, say

Derivative as a Linear Transformation - Differential Calculus, CSIR-NET Mathematical Sciences - Notes | Study Mathematics for IIT JAM, CSIR NET, UGC NET - Mathematics

By inspection, as it were, we can separate the right hand side into three parts. We have

Derivative as a Linear Transformation - Differential Calculus, CSIR-NET Mathematical Sciences - Notes | Study Mathematics for IIT JAM, CSIR NET, UGC NET - Mathematics

and the linear part of f is the vector

Derivative as a Linear Transformation - Differential Calculus, CSIR-NET Mathematical Sciences - Notes | Study Mathematics for IIT JAM, CSIR NET, UGC NET - Mathematics

which can be written in matrix form as

Derivative as a Linear Transformation - Differential Calculus, CSIR-NET Mathematical Sciences - Notes | Study Mathematics for IIT JAM, CSIR NET, UGC NET - Mathematics

By analogy with the one-dimensional case, we might guess that

f (x) = f (0) + Ax + an error term of order 2 in x; y:

where A is the matrix

Derivative as a Linear Transformation - Differential Calculus, CSIR-NET Mathematical Sciences - Notes | Study Mathematics for IIT JAM, CSIR NET, UGC NET - Mathematics

And this suggests the following

De nition: A function f : Rn → Rm is said to be differentiable at the point x = a ∈ Rn if there exists an m x n matrix A and a function ∈(x, a) such that

f (x) = f (a) + A(x - a) + ∈(x, a),

where

Derivative as a Linear Transformation - Differential Calculus, CSIR-NET Mathematical Sciences - Notes | Study Mathematics for IIT JAM, CSIR NET, UGC NET - Mathematics

The matrix A is called the derivative of f at x = a, and is denoted by Df (a).

Generalizing the one-dimensional case, it can be shown that if

Derivative as a Linear Transformation - Differential Calculus, CSIR-NET Mathematical Sciences - Notes | Study Mathematics for IIT JAM, CSIR NET, UGC NET - Mathematics

is differentiable at x = a, then the derivative of f is given by the m x n matrix of partial derivatives

Derivative as a Linear Transformation - Differential Calculus, CSIR-NET Mathematical Sciences - Notes | Study Mathematics for IIT JAM, CSIR NET, UGC NET - Mathematics

Conversely, if all the indicated partial derivatives exist and are continuous at x = a, then the approximation

Derivative as a Linear Transformation - Differential Calculus, CSIR-NET Mathematical Sciences - Notes | Study Mathematics for IIT JAM, CSIR NET, UGC NET - Mathematics

is accurate to the second order in x a.

Exercise: Find the derivative of the function f : R2 → R3 at a = (1, 2)t, where

Derivative as a Linear Transformation - Differential Calculus, CSIR-NET Mathematical Sciences - Notes | Study Mathematics for IIT JAM, CSIR NET, UGC NET - Mathematics

The document Derivative as a Linear Transformation - Differential Calculus, CSIR-NET Mathematical Sciences - Notes | Study Mathematics for IIT JAM, CSIR NET, UGC NET - Mathematics is a part of the Mathematics Course Mathematics for IIT JAM, CSIR NET, UGC NET.
All you need of Mathematics at this link: Mathematics
559 videos|198 docs
Use Code STAYHOME200 and get INR 200 additional OFF
Use Coupon Code
559 videos|198 docs
Download as PDF

Download free EduRev App

Track your progress, build streaks, highlight & save important lessons and more!

Related Searches

CSIR-NET Mathematical Sciences - Notes | Study Mathematics for IIT JAM

,

Summary

,

Objective type Questions

,

MCQs

,

UGC NET - Mathematics

,

Derivative as a Linear Transformation - Differential Calculus

,

Previous Year Questions with Solutions

,

study material

,

practice quizzes

,

ppt

,

Important questions

,

Derivative as a Linear Transformation - Differential Calculus

,

CSIR-NET Mathematical Sciences - Notes | Study Mathematics for IIT JAM

,

CSIR NET

,

shortcuts and tricks

,

CSIR-NET Mathematical Sciences - Notes | Study Mathematics for IIT JAM

,

Extra Questions

,

Derivative as a Linear Transformation - Differential Calculus

,

Viva Questions

,

mock tests for examination

,

CSIR NET

,

CSIR NET

,

Exam

,

Sample Paper

,

UGC NET - Mathematics

,

pdf

,

UGC NET - Mathematics

,

past year papers

,

video lectures

,

Semester Notes

,

Free

;