How Do Total and Partial Derivatives Differ in Multivariable Calculus?

[Hypnotoad]

Is there some underlying difference between the two types of derivatives?Other than the obvious that one is used on single variable functions, while the other is for multivariable functions. I'm asking because my classical professor mentioned something about knowing the difference between the two and how it was important for using the Hamiltonian and getting the equations of motion. I haven't taken a calculas class in about 5 years, and I don't remember any discussion on other differences. Thanks in advance for the help.

 

[Tide]

Consider that it is possible to calculate the total derivative of a function of several variables.

 

[wais]

Yes, there is an underlying difference between total and partial derivatives. While both types of derivatives involve finding the rate of change of a function, they differ in terms of the variables that are held constant during the differentiation process.

In total derivatives, all variables in the function are allowed to vary, and the derivative is taken with respect to a different variable. This means that the resulting derivative takes into account the changes in all variables, not just the one being differentiated with respect to. This is often used in single variable functions to find the instantaneous rate of change.

In contrast, partial derivatives only consider the changes in the function with respect to one variable, while holding all other variables constant. This is useful in multivariable functions, where it is not possible to take the derivative with respect to all variables at once. Instead, partial derivatives allow us to analyze how the function changes with respect to each individual variable.

The concept of partial derivatives is crucial in many areas of physics and mathematics, including the Hamiltonian and equations of motion that your professor mentioned. In these cases, we need to consider how a function changes with respect to each variable separately, and partial derivatives allow us to do so.

In summary, while both total and partial derivatives involve finding the rate of change of a function, they differ in terms of the variables that are held constant during the differentiation process. Understanding this difference is important in various mathematical and scientific applications.