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I am reading this document:

https://people.math.gatech.edu/~cain/notes/ta.PDF

I made a screenshot of the part of the document that I would like to ask my question on:

enter image description here

I am more or less familiar with the concept of composite function and derivative of such functions using the chain rule. This document mentions it. What I don't understand is how we go from the function $g(t)$ defined as:

$g(t) = f(x(t), y(t), z(t))$

to the derivative of this function defined as:

$g'(t) = \dfrac {\partial f} {\partial x} \bigg\rvert_{P_0}\dfrac {d x} {d t} \bigg\rvert_{t_0} + \dfrac {\partial f} {\partial y} \bigg\rvert_{P_0}\dfrac {d y} {d t} \bigg\rvert_{t_0} + \dfrac {\partial f} {\partial z} \bigg\rvert_{P_0}\dfrac {d z} {d t} \bigg\rvert_{t_0}$

More specifically:

  1. why do we use the notation $\dfrac {\partial f} {\partial z}$ and $\dfrac {d x} {d t}$? Why using $\partial$ in one case and $d$ in the other? Why not $\partial$ or $d$ in both cases. Is there a reason for that?
  2. why is the derivative a sum of these terms?

1 Answers 1

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This comes from the idea of total derivative versus partial derivative. In words, the total change in a function is equal to the sum of the partial changes. In general, this is related to the Jacobian of the function defined by

$J_{ij}=\frac{\partial f_i}{\partial x_j}$

which satisfies

$f(x+\delta x)-f(x)=J\delta x,$

for $\delta x$ very small. This is a local approximation for the function $f$. The $\partial$ terminology means taking the derivative with all other variables fixed, whereas the $d$ terminology means the total change in the function with respect to a given change in the underlying variables. Because x,y and z are functions of one variable (t), you can use the total derivative, whereas g depends on multiple variables, so you need to have the full equation written down.

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    Great fantastic, do you know where I could learn more about this? A pointer? I will google it of course, but if you can point me to something where things are clearly explained I would appreciate it (I am taking about the derivative versus partial derivative topic and the Jacobian of the function). Thx again – 2017-01-13
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    I think this is basically multivariable calculus, and also requires some linear algebra. I don't know a good book on those subjects, but most should do, also there is this https://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/. If you want to go deeper into this, I would recommend learning some differential geometry, maybe Differential Geometry of Curves and Surfaces by Do Carmo, which is a good book. – 2017-01-13
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    A more rigorous approach can also be found in books on real analysis, such as the one by John M. Howie. – 2017-01-13
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    One last question) then you answered my question 1) and 2) though am I right to think that the product of the terms is the result of the chain rule? (in g'(t)). In other words $g'(t) = r'(t) \cdot f'(g(t))$ and then you apply the approximation for the total derivative you are talking about to arrive to the final formula. Is this the right thinking? – 2017-01-13
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    That is more or less right but I think you made a typo - g(t)=f(r(t)) so you use the chain rule for that. Also this is the chain rule in higher dimensions, which is why you get a dot product. The chain rule in higher dimensions is related to these total derivative things, see the wikipedia article for chain rule under higher dimensions. – 2017-01-15