multivariate calculus -chain rule in partial differentiation

Question

Solving PDE equations I realized that I am a bit confused with it the chain rule and that I have go back to basics. Maybe it is just a notation thing.

For example
1) for one variable $$y=f(x), \ \ x=g(t) \implies f=( \ ( g(t) \ ) $$ $$\frac{dy}{dt} = \frac{dy}{dx} \frac{dx}{dt} \quad \quad \quad (1)$$

2) for two variables $$z=f(x,y), \ x=g(t), \ y=h(t) \implies z=f(g(t),h(t))$$

$$\mathbf{\frac{dz}{dt}} = \frac{\partial f}{\partial x} \mathbf{\frac{d x}{d t}} + \frac{\partial f}{\partial y} \mathbf{\frac{d y}{d t}} \quad \quad \quad (2)$$ $$$$

$$\mathbf{\frac{dz}{dt}} = \frac{\partial f}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial t} \quad \quad \quad (3)$$ $$$$ $$\frac{\partial z}{\partial t} = \frac{\partial f}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial t} \quad \quad \quad (4)$$ $$$$

Maybe it just a thing of notation convention when in case of one variable $\frac{\partial x}{\partial t}=\frac{dx}{dt}$ (is it correct?)

Can anybody clarify the differences between 2, 3 and 4?

Answer 1

You are correct:

• If $v$ is a function of one variable (call it $t$) you should use $dv/dt$ or $v'(t)$.
• If $v$ is a function of more than one variable (one of them being $t$) you should use $\partial v/\partial t$, or $v_t$ (or one of the other notations for a partial dervivative).

Equation (2) uses correct notation. It might help to instead write $$z(t)=f((g(t),h(t))$$

Then the derivative of $z$ is (suppressing the argument of the partial derivatives $f_1$ and $f_2$):

$$z'(t)=f_1g'(t)+f_2h'(t)$$

or using Leibniz notation:

$$\frac{dz(t)}{dt}=\frac{\partial f}{\partial x}\frac{dg(t)}{dt}+\frac{\partial f}{\partial y}\frac{dh(t)}{dt}$$

Alternatively you can write $dx/dt$ instead of $dg/dt$ and $dy/dt$ instead of $dh/dt$:

$$\frac{dz(t)}{dt}=\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt}$$

which is your equation (2).

Your equation (3) uses $\partial x/\partial t$, which is wrong because $x$ is only a function of one variable ($t$). Similarly, equation (4) uses $\partial z/\partial t$, which is the wrong notation because $z$ is a function of only one variable ($t$) (so long as you are thinking of $z$ as the composition of the function $f$ with the function $(g,h)$ rather than the function $f$ of two variables).