Total derivatives-how to get them, what they look like.

[vvarma]

I'm fairly confused as to how to get the total derivative of a function from Rⁿ to R^m. I know the definition and the meaning, i.e. best linear approximation to a function at given point, but my book, Rudin's Principles of Mathematical Analysis, is lacking in examples of how to actually get them. So when a problem asks me to get the derivative for some f(x), I'm invariably stumped.

I am wondering if anyone can possibly guide me through what I should think/do when faced w/ such a question. If something asks me to find the derivative, am I actually supposed to get the matrix of partials or proceed through the definition of total derivative and find the linear transformation that makes the error = 0.

For example, here are three problems from my last pset [the due date has passed so at this point I just want to understand it]:

Let all f:Rⁿ to R. Find Df at all points x in Rⁿ.
(a) f(x)=a dot x where a is fixed vector in Rⁿ.

This is by far the easiest since er(h) can be made exactly zero for Df(x)=a and I get the same thing from partials inserted into the jacobian matrix.

(b) f(x)=x dot L(x) where L(x) in a linear function from Rⁿ to Rⁿ.

I actually don't see what to do. Possibly the product rule?

(c) f(x)=||x||⁴ where || || is euclidean norm on Rⁿ.

I tried to make f=g² and then take the derivative but then the problem is again of how to take the derivative...Should I take partials? I can't see a way of doing that.

I'm sorry if this should be in another forum but I really am new to this place.

 

[snipez90]

b) Yes the (dot) product rule can be used. You should also know, at least intuitively, what the total derivative of a linear function is.

c) The way I might have done this is via http://en.wikipedia.org/wiki/Derivative#Directional_derivatives", which are more general than partial derivatives in that we care about the rate of change of f along any particular direction, not necessarily along the coordinate axes. It turns out that directional derivatives are fairly easy to compute with a certain lemma: Suppose the directional derivative of f in the direction v at x Dvf(x) exists. If F(t) = f(x + tv), then F'(0) = Dvf(x). This will allow you to compute the directional derivative of ||x||^4 without too much effort. Then there is a simple relationship between the total derivative and the directional derivative. Can you establish it?

The best advice I can give you though is to supplement Rudin with other multivariable calculus treatments. Apostol's Mathematical Analysis Ch. 12, Lang's Undergraduate Analysis, and Spivak's Calculus on Manifolds Ch.3 are good references. I assume you have access to a university library.

 

[vvarma]

snipez90: Thanks so much for the help.

(b) Using Df(x)=Dx dot L(x) + x dot DL(x) gives me Df=2(x dot L(x)) since derivatives of linear functions are the linear functions themselves and x in above map can be called Id(x) which, along w/ L(x), is linear. Is that possibly correct?

(c) The relationship being that if the partial derivatives exist and are continuous, then the total derivative exists, is given by the Jacobian, and depends continuously on x? So then showing the directional derivatives existence for an arbitrary direction as you say (whilst also showing continuity) would get me the total deriv. which happens to be the directional derivative.

 

[Bacle]

The total derivative is the best local linear approximation to the change of your
function. The Jacobian Df describes the linear map with respect to the standard
basis.