Stanton D. answered • 03/24/22
Tutor to Pique Your Sciences Interest
So Ruveshan N.,
First, write the function for the body diagonal length L in terms of x, y, and z. If you don't remember the Pythagorean Theorem (?!), look it up. Now you are going to take the derivative of that function with respect to the partial derivatives d/dx, d/dy, and d/dz . No problem, for each partial derivative, treat the other dimension variables as constants. Then add the 3 partials together.
Note some things in advance: when a particular dimension is changing, the rate of change of that one goes into its partial calculation, but the instantaneous values of the other variables go in.
Note also that when you wrote the expressions for each partial derivative (they look identical, except for the variables, don't they!), you automatically took into account the Pythagorean-Theorem way that they add up. So the sum of the 3 partials is a "scalar" for L, even though the movement of the free corner (assuming the diagonal corner is anchored at the origin) is a vector. Curious the way that works, isn't it. But that's the beauty of mathematics -- it makes difficult problems into easy ones to solve!
Hope that gives you a firm footing, as well as some perspective, on such problems.
-- Cheers, -Mr. d.
Stanton D.
Oh and by the way -- the above process works fine for derivatives. BUT, as you might imagine, the reverse process -- integrating to find the volume of an arbitrary rectangular prism, given polynomial functions in the "other" two variables for each partial derivative, is quite a more complex operation!03/24/22