## The Geometry of the Derivative for Real Valued Mappings (Pt. I)

**Point of Post: **In this post I’d like to discuss some of the geometric aspects of what the total and partial derivatives mean including the idea of approximating lines and tangent planes.

*Motivation*

As usual in math it’s helpful to have a picture to backup the ideas. In this post we discuss what it geometrically looks like when a mapping is differentiable at a point in terms of tangent planes. This of course generalize the notion that a mapping is differentiable at a point if it has a tangent line there.

*Tangent Planes*

Recall from basic geometry that a hyperplane in is a set of vectors which are mutually orthogonal to some fixed vector, call the *normal*, which pass through some specified point. In particular, if with we define the *hyperplane through with normal *, denoted , to be the set . If we call a hyperplane just a *plane*. Of course a hyperplane is just an -dimensional subspace of which is shifted by a certain vector. Indeed, if one defines the linear functional by then we know that is a -dimensional subspace of and it’s easy to see that . Said in more common language we see that . Thus, we can alternatively define as if one likes more explicit formulae.

Let , we say that the hyperplane is *tangent to at * if and for any sequence in with . Note that if we use the common definition of angle in given by then tangency is equivalent to saying that for every which clearly satisfies our intuitive notion of tangency.

What we now show is that every differentiable function possesses a tangent plane. But first, we recall the definition of the graph of a function is defined by . In what follows we shall think of the graph of a mapping as sitting inside instead of in the obvious way.

**Theorem: ***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 ).*

**Proof: **Let be a sequence in , then by definition with (this is enough since we know that must be continuous) Thus, we get that

but this last term goes to zero by definition of the derivative. Since the sequence was arbitrary the conclusion follows.

For the sake of pedagogy let us do an actual example. Take to be the hyperboloid . Clearly then is smooth on all of . Let us then compute the tangent plane to at . In particular since and and so the tangent plane is equal to or when one writes it out it is given by the equation . Graphing this gives us the following picture:

Evidently from the above theorem we recover the formula taught in every multivariable calculus course. Namely, if , with open, is differentiable at then the tangent plane to at is given by the equation

**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.

**Photo Credit:**

No comments yet.

## Leave a Reply