Partial Differentiation, complication in variables held constant

[maple]

Hi,

this is a pretty trivial question. would be grateful if someone could answer it for me.

in spherical polars x=rcos(theta)sin(PHI) and so on for y, and z

Now, why is

d/dr= dx/dr*d/dx + dy/dr*d/dy+ dz/dr*d/dz

where everything is partial. dx/dr, dy/dr and dz/dr at partial derivates held at contant thetha and phi.

why are they held at constant thetha and phi?

r^2=x^2 + y^2 + z^2

so r=function of (x,y,z)

thus we can write this out as an exact differential we get:

dr=dr/dx*dr + dr/dy*dy + dr/dz*dz

dr/dx is held constant wrt y,z etc. and NOT thetha and phi.

Can some explain how the differential at the top works.

thanks

 

[dextercioby]

"constant theta & phi"...It's nothing special,really.It's the plain old CHAIN RULE for partial derivatives.An implicit dependence of variables requires this chain rule...U should write

$$f\left(x(r,\vartheta,\varphi),y(r,\vartheta,\varphi),z(r,\vartheta)\right)$$

and the simply apply the chain rule...

Partial differentiation is ordinary differentiation,but with some of the variables "kept constant" while evaluating that limit implied by the def.

Daniel.

 

[maple]

thanks daniel,

can you actually spell all this out? I'm not sure how to put what you said into practice.

Many thanks.

 

[dextercioby]

"f" depends on $r$ implicitely,by means of the functions $x(r,...),y(r,...),z(r,...)$.So applying the chain rule,one gets simply

$$\frac{\partial f}{\partial r} =\frac{\partial f}{\partial x}\frac{\partial x}{\partial r} + \frac{\partial f}{\partial y}\frac{\partial y}{\partial r}+\frac{\partial f}{\partial z}\frac{\partial z}{\partial r}$$

and the same for the other 2 implicit variables.

U can use the diffeomorphism

$$\left \{ \begin{array}{c} x=r\sin\vartheta \cos\varphi \\ y=r\sin\vartheta \sin\varphi \\ z=r\cos\vartheta$$

together with

$$f\left(x\left(r,\vartheta,\varphi\right),y\left(r,\vartheta,\varphi\right),z\left(r,\vartheta\right)\right) = x^{2}+z^{3}y+\sqrt{xyz}$$

to find the 3 partial derivatives wrt the implicit variables...


Daniel.