Question

in multivariable calc, i dont understand why the chain rule is designed for us to take the derivative in terms of one variable t. Why do say z=xy and then x=g(t) and y=f(t). What thats doing is just rewriting z in terms of one variable and i dont understand what the utility of that is if we care about z being a function of two variable, x,y and not only one, t.

Answer 1

do you know what i mean?

Answer 2

The utility is that it's often a lot easier to work with a single variable than multiple variables. Even if we do care about the multiple variable form, by parameterizing the equation we might be able to understand the equation better.

Answer 3

wait so the answer is in terms of t and we usually leave it in terms of t, right? So are you saying that that answer is a parametric curve?

Answer 4

a curve in a 3d system... thats just confusing...

Answer 5

I usually try not to visualize the actual curve :P

But yes. Whenever you change your variables, you're finding a parametric curve.

Answer 6

ok thinking about it visually is confusing. So with the answer in terms of t, you can later say: "at t=3, the partial derivative in terms of x at that point is ..." right?

Answer 7

Yes. You can later say that. As long as your careful with the chain rule, everything works out fine.

Answer 8

could you figure out what the cartesian coordinates are t=3 too?

Answer 9

To get the cartesian coordinates, you would need to go back to the x,y, and z coordinate system by carefully applying the chain rule and doing some substitutions.

Answer 10

oh i see! going back to the beginning, if you change the multivariable function and you parametrize it to make it all in terms of , dont you fundamentally change the function... because it used to be a 3d surface and you reduce it to just a simple 2d curve?

Answer 11

*in terms of t*

Answer 12

Remember, not all functions can be easily parameterized to only a single variable t. Thos that can, are reduced to being a 2d curve. However, it's a 2d curve in a different coordinate system.

Answer 13

ok, you would to employ a vector parametric equation to sketch a curve along the surface in 3d?

Answer 14

am i getting my concepts right?

Answer 15

I'm not quite sure what you mean in the last question there.

Answer 16

if you want to make a 3d curve that "rides" along the 3d surface, would you have to use a vector equation?

Answer 17

and depending on how fast or where you want that curve to go through on the surface you would let t be different things...

Answer 18

That's basically what you have to do. You take the "gradient" of the 3d curve, and the gradient is a vector equation in three dimensions.

Answer 19

wait gradient.... does that have something to do with directional derivatives?
i learned about those in class just today

Answer 20

the directional derivative is a vector right

Answer 21

Correct. The gradient is the nest step of the directional derivatives. The gradient basically is\[a\vec{i}+b\vec{j}+c\vec{k}\]where a,b,c are the directional derivatives in the x,y, and z directions.

Answer 22

what do u mean by next step?

Answer 23

As in, you learn directional derivatives, and then you learn about the gradient.

Answer 24

ok

Answer 25

thanks !!

Answer 26

You're welcome.