Question

let W(s,t) =F(u(s,t))where F,u,v are differentiable and u(1,0) =2; v(1.0)=3;us(1,0)=-2;vs(1,0)=5;ut(1,0)=6;vt(1,0)=4;Fu(2,3)=-1;Fv(2,3)=10. Find Ws(1,0) and Wt(1,0)

Answer 1

it's quite easy to just put everything into the formula to get the answer is 34. my question is why dF/du at (1,0) can be calculate by Fu(2,3).the same problem with dF/dv

Answer 2

where is v in your formula?

Answer 3

F(u(s,t),v(s,t)). where... yes. it is

Answer 4

W(1,0)=F(u(1,0),v(1,0))and we are told that u(1,0)=2 and v=(1,0)=3, so
W(1,0)=F(2,3)

Answer 5

looks like multivariable chain rule problem ...

Answer 6

@experimentX yes, it is.
@TuringTest and then?

Answer 7

Fu(2,3) <--- this 2 and 3 are the values of u and v when (s,t) = (1,0) .. this is just change of variables.

Answer 8

you mean\[Ws = \frac{ \delta F }{ \delta u }\frac{ \delta u }{ \delta s }+\frac{ \delta F }{ \delta v }\frac{ \delta v }{ \delta s }\] and

Answer 9

\[\frac{ \delta F }{ \delta u }= F(2,3)\]

Answer 10

is it right?

Answer 11

the second formula you wrote is not right, the first is

Answer 12

\[ F_u(u(1,0), v(1,0))\]

Answer 13

@TuringTest mine is wrong? why?
@experimentX I think i got it. need time to make it clear in my mind. i will let you know when i get it perfectly. thanks anyway

Answer 14

you can't simply equate the partial differential with the function itself.

Answer 15

yes, i don't understand the goal of this part. we take partial derivative for what? tangent line respect to coordinates?

Answer 16

\[ \frac{ \partial F }{ \partial u }= F_u(2,3)\]
is the correct expression ... this does't have any goal.
this is simply like You know a function W of 's' and 't' ... but you do not know it as an explicit function of s and t.

Answer 17

instead you know it a function of 'u' and 'v' ... which in in turn function of 's' and 't' ... how do you find the tangent at given values of 's' and 't'?

Answer 18

Hey. think about t is time, s is the length, and v is velocity which function respect to s,t. and we need those part to get the function of v and some u(as a variable which respect to s, t , too. I don't think we study for nothing . if putting everything in big scenario, i think we get more effect than just formula, formula...and formula

Answer 19

suppose ... we have function like this,
\[ \huge W(s,t) = \int_0^s \cosh (tx) dx + \sin(s)\cos(t) \]
how do you calculate those partial values

Answer 20

ok, you are right, antiderivative for those parts? we must know the technique to solve that problem, but it must be used in somewhere, right? to me, the goal of partial derivative is find out the slope of tangent line in R3 respect to x,y,z.

Answer 21

yeah ... it is much easier to solve it (for partial) ... we can solve for it without knowing how to evaluate that integral.

Answer 22

to x axis, the tangent line has the slope totally different from it is to y axis and z axis as well. and those tangent line has the different equation, too. and then, to each dimension in R3, each of them respect to time, to velocity of something or at least to a stable origin as earth

Answer 23

ok, sorry, I think tooooo far. waste your time

Answer 24

its okay ... the partial with respect to time will give you velocity along x and y axis

Answer 25

yeah. when i were at kindergarden school, study addition and subtraction to count money or count my fingers. a little bit older, study multiplication and division to know how to count money and account. now, study partial derivative for ,,,,,don't know is really boring.

Answer 26

i've been doing this for last 20 years

Answer 27

and never ask yourself the silly question like me?

Answer 28

of course i've in the past .. i think you are just little confused with the notation. while doing these sorta question .. it does smoothly unless it's hard type.

Answer 29

Thanks for answering my question. I have many, many many homework from my classes. i wish i have 28 hours a day to solve all of them and to completely understand all of them, Thanks a lot. I have to go to school now. Hopefully I can get your help whenever I stuck at somewhere

Answer 30

sure ... anytime!!

Answer 31

Bye bye.