Engineering Calculus Notes 259

Engineering - 247 3.3 DERIVATIVES Finally we note that as a corollary of Proposition 3.3.7 we get a formula for the partial derivatives of the

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Unformatted text preview: 247 3.3. DERIVATIVES Finally, we note that, as a corollary of Proposition 3.3.7, we get a formula for the partial derivatives of the composite function g ◦ f : ∂f − ∂g ◦ f − (→0 ) = g ′ (y0 ) x (→0 ) . x ∂xi ∂xi For example, suppose we consider the function that expresses the rectangular coordinate y in terms of spherical coordinates: f (ρ, φ, θ ) = ρ sin φ sin θ ; its gradient is → − ∇ f (ρ, φ, θ ) = (sin φ sin θ, ρ cos φ sin θ, ρ sin φ cos θ ). Suppose further that we are interested in z = g (y ) = ln y. To calculate the partial derivatives of z with respect to the spherical coordinates when ρ=2 π φ= 4 π θ= 3 we calculate the value and gradient of f at this point: ππ f 2, , 43 → − ππ ∇ f 2, , 43 = (2) 1 √ 2 1 () 2 1 =√ 2 √√ 3 31 √ ,√ ,√ = 22 22 (3.11) ...
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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