Math Help

The Multivariable Chain Rule

Example: Let w = f( r ) where r = r( x, y, z) = √ [x² + y² + z² ]

Here w is the dependent variable, r is the intermediate variable, and

x, y, and z are the independent variables

Show that ∂²w/ ∂x² + ∂²w/ ∂y² + ∂²w/ ∂z² = d²w/ dr² + [2/r][dw/ dr]

The tree structure for this problem is as follows:

Start with the first derivative.

∂w/ ∂x = [dw/dr] [∂r/ ∂x] = dw/dr (1/2) [x² + y² + z² ]^-1/2[2x]

So ∂w/ ∂x = [ x/ √ [x² + y² + z² ] dw/dr = (x/r) dw/dr

Similarly, ∂w/ ∂y = [ y/ r ] dw/dr and ∂w/ ∂z = [ z/ r ] dw/dr

Next find the second derivatives. This is the hard part of the calculation. The second

derivative is just the derivative of the first derivative. Use the result for ∂w/ ∂x .

∂²w/ ∂x² = ∂/∂x [∂w/ ∂x] = ∂/∂x { dw/dr [∂r/ ∂x] }

Next use the product rule to obtain

∂²w/ ∂x² = ∂/∂x{dw/dr} [∂r/ ∂x] + dw/dr ∂/∂x {∂r/ ∂x}

Next interchange the order of integration for the first term on rhs which gives

∂²w/ ∂x² = d/dr{∂w/ ∂x } [∂r/ ∂x] + dw/dr ∂/∂x {∂r/ ∂x}

Now ∂w/ ∂x = [dw/dr] [∂r/ ∂x]

Perform the differentiations on the rhs (right hand side) gives

∂²w/ ∂x² = d²w/dr² [∂r/ ∂x]² + dw/dr [∂²r/ ∂x²]

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