Continuous partials implies differentiable - Revision history

Vipul: /* Related facts */

2012-05-12T16:42:18Z

Related facts

← Older revision		Revision as of 16:42, 12 May 2012
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	==Related facts==		==Related facts==

			===Generalizations===

			* [[Continuous partials of a given order implies differentiable that many times]]
	===Similar facts===		===Similar facts===

Vipul at 16:41, 12 May 2012

2012-05-12T16:41:51Z

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 <math>(\nabla f)(a_1,a_2,\dots,a_n) = \langle f_{x_1}(a_1,a_2,\dots,a_n), f_{x_2}(a_1,a_2,\dots,a_n), \dots, f_{x_n}(a_1,a_2,\dots,a_n) \rangle</math>
 ==Significance==
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 That is where the result of this page comes in useful. It allows us to use the continuity of the partial derivatives, that we have already computed, to deduce the existence of the gradient vector, and hence also compute an expression for it.

Vipul: Created page with "==Statement== Suppose $f$ is a real-valued function of $n$ variables $x_1,x_2,\dots,x_n$ . Suppose that $(a_1,a_2,\dots,a_n)$ is a ..."

2012-05-12T16:37:43Z

Created page with "==Statement== Suppose <math>f</math> is a real-valued function of <math>n</math> variables <math>x_1,x_2,\dots,x_n</math>. Suppose that <math>(a_1,a_2,\dots,a_n)</math> is a ..."

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==Statement==

Suppose <math>f</math> is a real-valued function of <math>n</math> variables <math>x_1,x_2,\dots,x_n</math>. Suppose that <math>(a_1,a_2,\dots,a_n)</math> is a point in the domain of <math>f</math> such that the [[partial derivative]]s <math>f_{x_1}, f_{x_2},\dots,f_{x_n}</math> exist and are continuous at and around the point <math>(a_1,a_2,\dots,a_n)</math> (i.e., they all exist and are continuous in an open ball containing <math>(a_1,a_2,\dots,a_n)</math>). Then, the [[gradient vector]] of <math>f</math> exists at <math>(a_1,a_2,\dots,a_n)</math> and is given by (as per [[relation between gradient vector and partial derivatives]]):

<math>(\nabla f)(a_1,a_2,\dots,a_n) = \langle f_{x_1}(a_1,a_2,\dots,a_n), f_{x_2}(a_1,a_2,\dots,a_n), \dots, f_{x_n}(a_1,a_2,\dots,a_n) \rangle</math>

==Significance==

In general, computing partial derivatives is easy, but computing the [[gradient vector]] from first principles is hard. Fortunately, we have a result, called the [[relation between gradient vector and partial derivatives]], that says that the coordinates of the gradient vector, wherever it does exist, are simply the partial derivatives. Explicitly, it says that:

<math>(\nabla f)(a_1,a_2,\dots,a_n) = \langle f_{x_1}(a_1,a_2,\dots,a_n), f_{x_2}(a_1,a_2,\dots,a_n), \dots, f_{x_n}(a_1,a_2,\dots,a_n) \rangle</math>

The problem with that result is that it holds [[concept of equality conditional to existence of one side|conditional to the existence of the gradient vector]]. In fact, [[existence of partial derivatives not implies differentiable]]. Thus, after computing the partial derivatives, we know what the gradient vector should be ''if it exists'', but we still aren't sure whether it exists.

That is where the result of this page comes in useful. It allows us to use the continuity of the partial derivatives, that we have already computed, to deduce the existence of the gradient vector, and hence also compute an expression for it.

Continuous partials implies differentiable - Revision history

Vipul: /* Related facts */

Vipul at 16:41, 12 May 2012

Vipul: Created page with "==Statement== Suppose f is a real-valued function of n variables x_1,x_2,\dots,x_n. Suppose that (a_1,a_2,\dots,a_n) is a ..."

Vipul: Created page with "==Statement== Suppose $f$ is a real-valued function of $n$ variables $x_1,x_2,\dots,x_n$ . Suppose that $(a_1,a_2,\dots,a_n)$ is a ..."