## Directional Derivatives and Partial Derivatives

**Point of Post: **In this post we discuss the notions of directional derivatives and partial derivatives

*Motivation*

Roughly what the total derivative does is describe conditions when a function can be locally approximated very well (sublinearly) well by an affine transformation. Indeed, suppose that is differentiable at . By definition the limit for any we may choose such that implies . Note that that we see from this that ‘locally’ here means in all possible directions (as soon as is within the open ball the above inequality applies). Sometimes though we are only interested in the approximation, or notion of change in a particular direction. Thus arises the directional derivative which, roughly put, measures the rate of change of a function at a point towards a vector . The idea is simple, namely one takes, along the line from to , successive differences of the value and and let tend to zero. When the vector is one of the elements of the canonical basis we get the partial derivatives which, as we shall see, are of huge importance in multivariable differential analysis.

*Directional Derivatives*

Let be open and suppose that . Suppose that and , if the limit

(where the limit is taken over ) exists we say that has a *directional derivative at in the direction of *and denote this limit as (remembering that it’s the directional derivative of at in the direction of –syntactically the order of the symbols makes sense). It is important to note that this limit lives inside and that is open for similar convexity reasons as discussed for the total derivative. If is a canonical basis vector then we call the *partial derivative with respect to *or the *partial derivative with respect to the variable *and, when desirable, can alternatively denote it as (omitting parentheses around the since, thinking of as an operator, one usually writes and not ) or if we are thinking of our function as (i.e. the variables are denoted ). If instead we had a function and our variables are of the form we would denote as . If our function has partial derivatives in the direction of for every element of some region then we have a function .

If in the above definition (i.e. if is real-valued) then directional derivatives have a very nice interpretation. To get the idea we first inspect partial derivatives. Namely, since is open we can find some open ball . Then, we know that is contained in . We can then define a function given by . It’s not hard to see then that exists if and only if is differentiable (in the usual one-dimensional sense) at and moreover . This is really nice because it allows us to evaluate derivatives in the ‘everything else is constant’ way we learned in calculus. Namely, to find one merely ‘pretends’ the variables are constant in and differentiate with respect to normally. So that for example if then . Now, we could have phrased this a little less explicitly by noting that really . We can then generalize this to note that for any direction we could have considered (where lives in an interval defined analogously to above) and then realized that having a directional derivative in the direction of is equivalent to having be differentiable at and moreover that .

The first thing we’d like to show is that, just as for the total derivative, having a directional derivative in some direction is equivalent to having a directional derivative for each of the coordinate functions. More explicitly:

**Theorem: ***Let , where is open, and let . Then, has a directional derivative at in the direction of if and only if have directional derivatives at in the direction of and in which case .*

**Proof: **This follows immediately from the common theorem that a limit of a vector-valued function exists if and only if the limit of each coordinate function exists, and in which case the limit of the vector valued function is the tuple where each coordinate is the limit of the corresponding coordinate function.

We next prove that directional derivatives (and in particular partial derivatives) are’ homogeneous in the direction’ in the following sense:

**Theorem: ***Let , open, have a directional derivative at in the direction . Then, has a directional derivative at in the direction of for any and . *

**Proof: **To prove the first assertion we merely note from basic analysis that since we assumed exists that

exists and is equal to

This proves the intuitive idea that the directional derivative in the opposite direction should be the negative of the directional derivative in the positive direction.

We lastly note the following theorem whose proof is so obvious we omit it:

**Theorem: ***Let latex U\subseteq\mathbb{R}^n$ open, have directional derivatives at in the direction of . Then, for any constants one has that has a directional derivative at in the direction of and .*

**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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In the definition of directional derivative, the Wolfram site says u must be a unit vector. I’ve seen posts in math forums that complain about Apostol’s definition omitting that requirement.

Comment by Stephen Tashiro | September 2, 2011 |

Hello Stephen! I would be tempted to disagree with people’s objections. I mean really, it seems like a fairly artificial decision to restrict to unit vectors except that they form, in a sense, a representative class of vectors. Is there a reason you disagree with it?

Thanks for reading my blog!

Best,

Alex

Comment by Alex Youcis | September 3, 2011 |

I agree that Apostol’s definition is useful and more general that requiring that v be a unit vector. It bothers me a little that we only use the word “directional” for it. That doesn’t convey the “magnitude” part of the “magnitude and direction” of a vector.

Comment by Stephen Tashiro | September 3, 2011 |

I can see what you’re saying. Perhaps the best way to do it would be to define the directional derivative for arbitrary vectors and then discuss why it’s named what it is by considering the case when the vector is a unit.

Comment by Alex Youcis | September 3, 2011 |

In a thread on physicsforums.com, the poster Hootenanny says:

“Actually, upon re-reading Apostol’s definition as posted above, he does not mention the term “directional derivative”. It is Tomer that asserts that this is the directional derivative. Apostol only refers to the derivative with respect to a vector”

I don’t have a copy of Apostol’s book to check this.

Comment by Stephen Tashiro | September 6, 2011 |

Stephen,

Is there really a difference?

Comment by Alex Youcis | September 7, 2011 |

It’s just a technicality about terminology, not “real math”. However, if Apostol is the reference and he doesn’t refer to his definition as the definition of “directional derivative”, then I wouldn’t put those words in his mouth. Perhaps you have another reference that does call it a “directional derivative”.

Comment by Stephen Tashiro | September 7, 2011 |

Stephen,

Ah, I see what you mean now. I think the problem is what I mean by ‘reference’. There is a page on my blog that explains that by ‘references’ I really mean ‘place where you can find material similar to what I am doing’, so perhaps further reading. That said, I do on occasion write with a particular author in mind, and if so I try to make specific mention of this. Does that help clear things up?

Best,

Alex

Comment by Alex Youcis | September 7, 2011 |

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