Question

what is the geometric significance of taking partial derivatives?

Answer 1

basically, i know how to calculate them, but I do not understand what it acomplishes

Answer 2

to find out how does a function changes relative to one variable while keeping other ones fixed

Answer 3

idk, I understand how to do the math, but I do not understand what the math is for

Answer 4

Volume of cylinder

\[v= \pi r^2 h \]

how does volume change relative to change in radius while height remain contant?

Answer 5

as radius gets bigger, volume gets bigger?

Answer 6

Yeah, I mean more quantitatively.

Answer 7

to know that we take partial with respect to r

\[\partial v/dr=\pi h 2r\]

\[\partial v=\pi h 2r \space dr\]

Answer 8

right, i understand that, and a full derivative has something to do with tangents

Answer 9

dr = change in radius , Dv = change in radius

so if radius change by one , volume change by pi h 2 r

Answer 10

ok, i think i understand that, i keep forgetting that d stands for change, so back to my original question, the partial deriv. has to do with the change in the vector?

Answer 11

Say you have a three-dimensional space
a vector in that space has an x, y, and z component, all of which are perpendicular to each other.

Now, if I take the partial derivative of x with respect to y I am asking only about how much the x-component changes as I change the y-component.

That is why we treat z as a constant, we don't care about how much it changes in a partial derivative because we are looking at the changes in each individual component of the vector.

Does this help at all?

Answer 12

oh! ok. I think I get it now, I was trying to make it way more involved. Thank you :)

Answer 13

very welcome :)