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Total derivatives are calculated very similar to directional derivatives and I was wondering are they one and the same?

If not, what is the difference between them?

amWhy
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  • In the book "A Geometric Approach to Differential Forms" by David Bachman (Publisher: Birkhauser), Section 1.5 on the topic of Gradients is a very good, in fact, "excellent" development (with graphical depictions) of the Gradient and the Directional Derivative and how they are different. I highly recommend this book for anyone interested in this and similar topics. – K7PEH Nov 04 '14 at 16:02
  • related question: https://math.stackexchange.com/questions/1410308/is-the-total-differential-the-same-as-the-directional-derivative/2391377 – Felipe G. Nievinski Jul 14 '21 at 03:45

2 Answers2

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No it is not.

Total derivative;

$$ dF = \mbox{grad} \, F \cdot dx = \sum_i \partial_i F \, dx_i $$

Directional derivative:

$$ \frac{\partial F }{\partial n} = \mbox{grad} \, F \cdot n = \sum_i \partial_i F \, n_i $$

where $n$ is a unit vector which is used to specify the desired direction.

mvw
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  • could not $dx$ be a unit vector though? –  Nov 04 '14 at 15:08
  • I do not think so. $F$ is a linear combination of differentials, most likely to be used in some integration formula or some differential equation. $\partial F/\partial n$ is a common derivative. – mvw Nov 04 '14 at 15:11
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The total derivative you can think of as the collection of all the directional derivatives. If $F : \mathbb{R}^n \to \mathbb{R}^m$, and $D_a F : \mathbb{R}^n \to \mathbb{R}^m$ is its total derivative at point $a \in \mathbb{R}^n$ then $D_a F v$ is the directional derivative in the direction $v$.

So the total derivative, $D_a F$ is a machine that takes a direction, and gives you the corresponding directional derivative of $F$ at point $a$.

Sometimes it is convenient to take the direction to be a normal vector, but you could plug in vectors of any length.

The total derivative can be computed using the partial derivatives via the Jacobian.

  • They're the same thing, except total differentials are for functions $\mathbb{R}^n \to \mathbb{R}$ –  Nov 04 '14 at 15:20