Total derivative

Question

What is the significance and meaning of the total derivative? Why is it introduced in the definition of differentiability of scalar and vector fields?

Answer 1

When you are dealing with vectors, vector fields, and scalar fields, you often have more than one independent variable. The concept of partial derivatives becomes important. However, sometimes you will have a parametric representation which ultimately only depends on one variable, in which case you can find the total derivative with respect to that single independent variable using partial derivatives and the chain rule.

The total derivative of a function is the total rate of change.

For example, consider $f(t,x,y)$ where $x$ and $y$ are parametric functions of $t$: $x=x(t)$ and $y=y(t)$. The total derivative of $f$ w.r.t. $t$ is: $$ \frac{\text{d}}{\text{d}\ t}f(t,x,y)= \frac{\partial}{\partial t}f(t,x,y)+\frac{\partial}{\partial x}f(t,x,y)x'(t)+\frac{\partial}{\partial y}f(t,x,y)y'(t). $$

Even though $f$ is seemingly a function of three variables, it is really only a function of one variable, $t$.

Answer 2

I sometimes use the total derivative to solve differential equations. For example if you are trying to solve $\frac{dy}{dx} = -\frac{x}{y}$, it would be nice to recognize $x\ dx + y \ dy$ as the total derivative of $\frac{1}{2} x^2 + \frac{1}{2} y^2.$ now you can write down the solution, at least implicitly, as $x^2 + y^2 = \text{constant}.$

If it is not as simple as the example, you can try to make it into an exact equation, where the concept of the total derivative is useful for finding a solution. Any differential equation of the form $$M(x,y)dx + N(x,y)dy = 0$$ is exact if $$\frac{\partial}{\partial y}M(x,y)=\frac{\partial}{\partial x}N(x,y),$$ and can be solved by multiplying by an integrating factor. This works in some problems. Eventually you may end up solving a partial differential equation for the integrating factor.