Dr/dt = (dr/du)(du/dt) + (dr/dv)(dv/dt)

[jamesb1]

Sometimes I see the equation in the style as seen in the topic title (the d is the partial derivative) during vector analysis lectures and I get confused.

Its like saying dr/dt = dr/dt + dr/dt which doesn't make a lot of sense is it not?

 

[Solkar]

if $$r \equiv r(u,v)$$ then
$$dr = \frac{\partial r}{\partial u} du + \frac{\partial r}{\partial v} dv$$
is the total differential.

Divide that symbolically by dt.

 

[WannabeNewton]

Its like saying dr/dt = dr/dt + dr/dt which doesn't make a lot of sense is it not?

No it isn't and the reason is because you can't just divide out $\partial u$ and $du$ like that. The former is a nonsense expression. This is where the horrible practice of "dividing out" differentials from single variable calculus comes back to bite you. The expression in the title comes out of the chain rule.

For a proof of this chain rule, see here (Theorem 6): http://math.bard.edu/belk/math461/MultivariableCalculus.pdf

 

[jamesb1]

Wow I just realized I made a very silly misconception with regards to the partial and total derivatives. I fully understand now! Thank you for your help :)

 

[blue_raver22]

I can understand how this equation can be confusing at first glance. However, it is important to understand that the d in this equation represents the partial derivative, which is a mathematical concept used in multivariable calculus. In this context, the equation is stating that the rate of change of r with respect to time (dr/dt) is equal to the rate of change of r with respect to u (dr/du) multiplied by the rate of change of u with respect to time (du/dt), plus the rate of change of r with respect to v (dr/dv) multiplied by the rate of change of v with respect to time (dv/dt). This equation is known as the chain rule and is a fundamental concept in vector analysis.

So, while it may seem like the equation is stating that dr/dt is equal to itself, it is actually stating the relationship between the rate of change of r with respect to time and the rates of change of its individual components with respect to time. This equation is commonly used in vector analysis to calculate the overall rate of change of a vector with respect to time. I hope this explanation helps to clarify any confusion you may have had.