Chain Rule of Multivariable Calculus

[sams]

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...

 

[andrewkirk]

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.

 

[Stephen Tashi]

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.