Do these two partial derivatives equal each other?

[thegirl]

take the function f(x,y,z)

s.t dF=(d'f/d'x)dx+(d'f/d'y)dy+(d'f/d'z)dz=0 where "d'" denotes a curly derivative arrow to show partial derivatives
Mod note: Rewrote the equation above using LaTeX.
$$df = (\frac{\partial f}{\partial x} ) dx + (\frac{\partial f}{\partial y} ) dy + (\frac{\partial f}{\partial z} ) dz = 0$$

Is this statement true? (d'f/d'z)x=(d'f/d'z)y (the partial derivative of the function with respect to z at a constant x equal to the partial derivative of the function with respect to z with y as a constant?

Could someone explain this to me? Are those partial derivatives equal to each other?

 

[Mark44]

take the function f(x,y,z)

s.t dF=(d'f/d'x)dx+(d'f/d'y)dy+(d'f/d'z)dz=0 where "d'" denotes a curly derivative arrow to show partial derivatives
Mod note: Rewrote the equation above using LaTeX.
$$df = (\frac{\partial f}{\partial x} ) dx + (\frac{\partial f}{\partial y} ) dy + (\frac{\partial f}{\partial z} ) dz = 0$$

Note that I changed dF as you wrote it to df, assuming you're talking about the function f.

Is this statement true? (d'f/d'z)x=(d'f/d'z)y (the partial derivative of the function with respect to z at a constant x equal to the partial derivative of the function with respect to z with y as a constant?

Since df = 0, the f(x, y, z) = C, for some constant C.
$\frac{\partial f}{\partial z}$ is the partial derivative of f with respect to z. It's redundant to say the partial of f with respect to z, with x held constant or with y held constant. Since you're differentiating with respect to z, x and y are held constant anyway.

Could someone explain this to me? Are those partial derivatives equal to each other?

 

[Fredrik]

The notation that specifies which variables are held constant is only needed when it wouldn't be clear otherwise what function we're differentiating. For example, suppose that the 4 variables w,x,y,z satisfy the equations $x+y+z=1$ and $w=2x+y^2-z$. If you write $\frac{\partial w}{\partial x}$, it's not entirely clear if the function you want to differentiate is the g defined by $g(x,y)=2x+y^2-(1-y-x)$ or the h defined by $h(x,z)=2x+(1-x-z)^2-z$. If it's the former, you can write $D_1g(x,y)$ or $\left(\frac{\partial w}{\partial x}\right)_y$. If it's the latter, you can write $D_1h(x,z)$ or $\left(\frac{\partial w}{\partial x}\right)_z$.

The given equations define at least two functions (h and g) implicitly. The need to write out "what variable is held constant" arises when you chose not to introduce new symbols (like h and g) for those functions, and instead just use the symbol w for one of them.

 

[HallsofIvy]

You title this "Do these two partial derivatives equal each other" but what you have are three partial derivatives and what you are equating are differentials, not derivatives. Just as in single variable Calculus, we start by defining the "derivative" dy/dx= f'(x) but then define the "differentials" by "dy= f'(x)dx", so we define the "differential" for a function of several variables.

We can do this: Let f(x,y,z) be a function of the three variables x, y, and z, and suppose that each of x, y, and z is a function of some other variable, t. Then f can be thought of as a function of the single variable t and, by the chain rule, $\frac{df}{dt}= \frac{\partial f}{\partial x}\frac{dx}{dt}+ \frac{\partial f}{\partial y}\frac{dy}{dt}+ \frac{\partial f}{\partial z}\frac{dz}{dt}$. Now we can write that in "differential form" as $df= \frac{\partial f}{\partial x}dx+ \frac{\partial f}{\partial y}dy+ \frac{\partial f}{\partial z}dz$

 

[Fredrik]

You title this "Do these two partial derivatives equal each other" but what you have are three partial derivatives and what you are equating are differentials, not derivatives.

I think she was trying to ask if the partial derivative of f "with respect to z, and with x constant" is equal to the partial derivative of f "with respect to z, and with y constant". Of course, there's no need to say that one of the variables are held constant if f can only denote one function. The specification of what variable is held constant is meant to be the last piece of information you need to know what function f is. (See my previous post for some details).

This concept is often used in physics books (especially in thermodynamics), where it's convenient to have the same symbol for several different functions whose values are meant to be interpreted as values of the same physical quantity.