Chain Rule of Multivariable Calculus

  • #1
sams
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I am confused when I should use the notation and the d notation. For example, on http://tutorial.math.lamar.edu/Classes/CalcIII/ChainRule.aspx, in Case 1, the author wrote dz/dt while in Case 2, the author wrote ∂z/∂t. Could anyone please explain to me when I should use the notation and the d notation.

Any help is much appreciated. Thanks a lot...
 

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  • #2
andrewkirk
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In Case 1, z is regarded as a single-variable function of t, so the total derivative dz/dt is used.

In Case 2, z is regarded as a two-variable function of s and t, so derivatives wrt s and t are partial, and so are written ##\partial z/\partial t## and ##\partial z/\partial s##.

If z is a function of more than one variable, once the intermediate variables (x and y in this case) are eliminated, then the partial derivative notation is used, otherwise the total derivative notation.
 
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  • #3
Stephen Tashi
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There are two distinct meanings of taking the derivative of a function ##f(x,y)## "with respect to x".

These are

Case 1) Taking the partial derivative of ##f## with respect to its first argument, which has been named "x" in this example.

Case 2) Taking the dervative of ##f## with respect to the variable ##x## wherever ##x## is involved.

For example, suppose we are given
##f(x,y) = x^2 + y## and ##y = 3x^3##.

case 1) ##\frac{\partial f}{\partial x} = 2x##

case 2) ##\frac{df}{dx} = 2x + 9x^2##

These two distinct concepts are a perpetual source of confusion in reading material written by physicists. (See @andrewkirk 's https://www.physicsforums.com/insights/partial-differentiation-without-tears/ )

To avoid such confusion, some authors have gone so far as using notation like ##f_1(x,y)## when they refer to the derivative of ##f## with respect to its first argument. This is a more straightforward notation that traditional notation like ##\frac{\partial f}{\partial w}## because to know the meaning of that notation, you must know which position ##w## occupies in the list of arguments of ##f##.

The functions defined by ##f(w,r) = w^2 + 3r ## and ##f(r,w) = r^2 + 3w ## are the the same function unless other information has been given to distinguish ##w## and ##r##. The partial derivative of ##f## with repspect to its first argument can be denoted as ##f_1(x,y) = 2x## or ##f_1(w,r) = 2w## or ##f_1(r,w) = 2r##. Technically variable names are abitrary. In applying math, a variable talked about on one page may have a special meaning that carries over to the next page, so people don't exercise their full freedom of choice when defining functions. Also there are traditions such as using "##x##" to be the first argument of a function.
 
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