afafConsider f(r, y, 2) = 2² + y²z? and z = r sin 0, y = r cos 0, z = Or³. Findanddr

Answered: af ar fe Consider f(1, y, z) = r² +…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

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Consider f(r, y, 2) = 2² + y²z? and z = r sin 0, y = r cos 0, z = Or³. Find
and
dr

Expert Solution

Given:

The function $f (x, y, z) = x^{2} + y^{2} z^{2}$ and the polar transformation coordinates are $x = r \sin θ$ , $x = r \cos θ$ and $z = θ r^{3}$ .

Explanation:

Substitute the values $x = r \sin θ$ , $x = r \cos θ$ and $z = θ r^{3}$ in the function $f (x, y, z)$ .

$\begin{array}{rcl} f (r, θ) & = & r^{2} \sin^{2} θ + r^{2} \cos^{2} θ {(θ r^{3})}^{2} \\ = & r^{2} \sin^{2} θ + r^{2} \cos^{2} θ (θ^{2} r^{6}) \\ = & r^{2} \sin^{2} θ + θ^{2} r^{8} \cos^{2} θ \end{array}$

Obtain the derivative of the function $f (r, θ)$ with respect r.

$\begin{array}{rcl} \frac{\partial}{\partial r} (f (r, θ)) & = & \frac{\partial}{\partial r} (r^{2} \sin^{2} θ + θ^{2} r^{8} \cos^{2} θ) \\ = & 2 r \sin^{2} θ + 8 r^{7} θ^{2} \cos^{2} θ \end{array}$

Similarly, the derivative of the function $f (r, θ)$ with respect $θ$ .

$\begin{array}{rcl} \frac{\partial}{\partial θ} (f (r, θ)) & = & \frac{\partial}{\partial θ} (r^{2} \sin^{2} θ + θ^{2} r^{8} \cos^{2} θ) \\ = & r^{2} (2 \sin θ \cos θ) + 2 θ r^{8} \cos^{2} θ + (2 θ^{2} r^{8} \cos θ) (- \sin θ) \\ = & 2 r^{2} \sin θ \cos θ + 2 θ r^{8} \cos^{2} θ - 2 θ^{2} r^{8} \cos θ \sin θ \end{array}$

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ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated