Partial Differentiation, complication in variables held constant

  • Thread starter maple
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  • #1
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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
 

Answers and Replies

  • #2
dextercioby
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"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

[tex] f\left(x(r,\vartheta,\varphi),y(r,\vartheta,\varphi),z(r,\vartheta)\right) [/tex]

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.
 
  • #3
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thanks daniel,

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

Many thanks.
 
  • #4
dextercioby
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"f" depends on [itex] r [/itex] implicitely,by means of the functions [itex] x(r,...),y(r,...),z(r,...) [/itex].So applying the chain rule,one gets simply

[tex] \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} [/tex]

and the same for the other 2 implicit variables.

U can use the diffeomorphism

[tex] \left \{ \begin{array}{c} x=r\sin\vartheta \cos\varphi \\ y=r\sin\vartheta \sin\varphi \\ z=r\cos\vartheta [/tex]

together with

[tex] 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} [/tex]

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


Daniel.
 

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