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November 23, 2014

November 23, 2014

Posted by **Claire** on Saturday, January 9, 2010 at 8:05am.

2) If z = sin theta.sin phi.sin gamma, and z is calculated for the values theta = 30degrees, phi = 45 degrees and gamma = 60degrees, find approximately the change in the value of z if each of the angles theta and gamma is increased by the same small angle alpha degrees, and phi is decreased by 1/2 alpha degrees.

Can someone please show me how to work these out?

- Maths: Partial Differentiation -
**drwls**, Saturday, January 9, 2010 at 11:19am1) change your equations to

r^2 = x^2 + y^2

2r dr/dx = 2x

dr/dx = x/r = cos theta

cos theta = x/r

Now partially differentiate with respect to x.

-sin theta d theta/dx = 1/r

dtheta/dx = -1/(r sin theta) = -1/x

I had to use d for the slanted Greek partial symbol above

2) Assume alpha is very small and use

dz = (partialz/dtheta)*alpha + (partialz/dphi)*alpha/2 + (partialz/dgamma)*alpha

dz is the "total differential" increase in z.

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