Abstract Nonsense

Crushing one theorem at a time

Relationship Between the Notions of Directional and Total Derivatives (Pt.I)


Point of Post: In this post we show the relationship between total and directional derivatives, and in doing so finally find an explicit formula for the total derivative in terms of the partial derivatives.

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Motivation

So the question remains how the total derivative, which we said was a measure in-all-directions of local change and approximation to a function, and directional derivatives which we said was some kind of measure of change in a specified direction that ignored all others. Some things seem intuitively obvious. For example, one feels that functions which have total derivatives at a certain point should morally be obligated to possess directional derivatives in all directions. Moreover, it seems not too bizarre that the total derivative should be able to be expressed, in some way, by some combination of directional derivatives. I mean, it makes sense that the change in an arbitrary direction should have something to do with the way it’s “component” directions act. In fact, we shall prove both of these things–namely that total differentiability implies the existence of directional derivatives in all directions, and that the Jacobian can be expressed entirely in terms of partial derivatives. The surprising thing we shall show is that a fairly strong converse holds–namely that if all the partial derivatives are ‘nice’ (in a precise sense to define soon) then we are guaranteed total differentiability

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Total Differentiability and Directional Differentiability

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We begin by showing that total differentiability implies directional differentiability in a very interesting way. In particular:

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Theorem: Let f:U\to\mathbb{R}^m, where U\subseteq\mathbb{R}^n is open, be total differentiable at a\in U. Then, for every u\in \mathbb{R}^n we have that D_f(a;u), the directional derivative of f at a in the direction of u, exists and D_f(a;u)=D_f(a)(u).

Proof: By definition, since

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\displaystyle \lim_{h\to\bold{0}}\frac{\left\|f(a+h)-f(a)-D_f(a)(h)\right\|}{\|h\|}=0

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It’s not hard to see that this implies the much weaker result that

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\displaystyle \lim_{t\to0}\frac{\|f(a+tu)-f(a)-D_f(a)(tu)\|}{\|tu\|}=\bold{0}

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which clearly implies that

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\displaystyle \lim_{t\to0}\frac{f(a+tu)-f(a)-D_f(a)(tu)}{|t|}=0

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Thus, we have that

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\displaystyle \lim_{t\to0^\pm}\frac{f(a+tu)-f(a)-D_f(a)(tu)}{\pm t}=0

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(where t\to0^\pm is meant to encapsulate two limits–the limit t\to0^- and the limit t\to0^+)and so in either case

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\displaystyle \lim_{t\to0^\pm}\frac{f(a+tu)-f(a)}{t}=D_f(a)(u)

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and so the conclusion follows. \blacksquare

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As a corollary we get the following, very interesting theorem,

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Theorem: Let f:U\to\mathbb{R}^n, where U\subseteq\mathbb{R}^n is open, be differentiable at a\in U. Then, D_jf(a) exists for all j\in[n] and

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\text{Jac}_f(a)=\begin{pmatrix}D_1f_1(a) & \cdots & D_nf_1(a)\\ \vdots & \ddots & \vdots\\ D_1f_m(a) & \cdots & D_nf_m(a)\end{pmatrix}=\begin{pmatrix}\displaystyle \frac{\partial f_1}{\partial x_1}(a) & \cdots & \displaystyle \frac{\partial f_1}{\partial x_n}(a)\\ \vdots & \ddots & \vdots\\ \displaystyle \frac{\partial f_m}{\partial x_1}(a) & \cdots & \displaystyle \frac{\partial f_m}{\partial x_n}(a)\end{pmatrix}=\left[D_jf_i(a)\right]

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Proof: The existence of the partial derivatives follows from the previous theorem. The form of the Jacobian follows from the fact that by the previous theorem we know that D_f(a)(e_i)=D_if(a) and from previous theorem this is equal to \displaystyle \left(D_if_1(a),\cdots,D_if_m(a)\right)=\sum_{k=1}^{m}D_if_k(a)e_k from where the conclusion follows. \blacksquare

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In particular we see that if f:U\to\mathbb{R}, with U\subseteq\mathbb{R}^n, then really D_f(a)(x)=\left\langle x_0,x\right\rangle=x_0\cdot x (where \langle\cdot,\cdot\rangle and \cdot denote the usual inner product on \mathbb{R}^n) where x_0=\left(D_1f(a),\cdots,D_nf(a)\right). This vector is so important that it’s given a name. In particular, if f:\mathbb{R}^n\to\mathbb{R} then the vector \left(D_1f(a),\cdots,D_nf(a)\right) is called the gradient of f and is denoted \nabla f(a). Thus, in particular we see that if f:U\to\mathbb{R} then D_f(a;u)=\nabla f(a)\cdot u. Clearly if \nabla f(a) is defined for some open subregion R\subseteq U then we have the function \nabla f:R\to\mathbb{R}^n which shall be important later.

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References:

1.  Spivak, Michael. Calculus on Manifolds; a Modern Approach to Classical Theorems of Advanced Calculus. New York: W.A. Benjamin, 1965. Print.

2. Apostol, Tom M. Mathematical Analysis. Reading, MA: Addison-Wesley Pub., 1974. Print.

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June 2, 2011 - Posted by | Analysis | , , , , ,

2 Comments »

  1. […] Let , with open, be differentiable at . Then the hyperplane is tangent to at where is the gradient of (where we, as stated before this problem, freely identify with and with […]

    Pingback by The Geometry of the Derivative for Real Valued Mappings (Pt. I) « Abstract Nonsense | June 9, 2011 | Reply

  2. […] where is open, the point a regular value if for every with one has that is non-zero, which, recalling that , is equivalent to . With this in mind we can state the […]

    Pingback by Surfaces (Pt. II) « Abstract Nonsense | October 9, 2011 | Reply


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