Chapter 13: Q. 51 (page 1080)

The formulas for converting from cylindrical coordinates to rectangular coordinates are x = r cos θ, y = r sin θ, and z = z. Prove that the Jacobian $\frac{\partial (x, y, z)}{\partial (r, θ, z)} = r$ .

Short Answer

Expert verified

It is proven that $\frac{\partial (x, y, z)}{\partial (r, θ, z)} = r$

Step by step solution

fgjf

gfh

Given information

The transformation equations are,

$x = r \cos θ, y = r \sin θ, z = z$

Proof

The definition of Jacobian of a transformation using partial derivatives is given as

$L . H . S = \frac{\partial (x, y, z)}{\partial (r, θ, z)} \frac{\partial (x, y, z)}{\partial (r, θ, z)} = |\begin{matrix} \frac{\partial x}{\partial r} & \frac{\partial y}{\partial r} & \frac{\partial z}{\partial r} \\ \frac{\partial x}{\partial θ} & \frac{\partial y}{\partial θ} & \frac{\partial z}{\partial θ} \\ \frac{\partial x}{\partial z} & \frac{\partial y}{\partial z} & \frac{\partial z}{\partial z} \end{matrix}| \frac{\partial (x, y, z)}{\partial (r, θ, z)} = |\begin{matrix} \cos θ & \sin θ & 0 \\ - r \sin θ & r \cos θ & 0 \\ 0 & 0 & 1 \end{matrix}| \frac{\partial (x, y, z)}{\partial (r, θ, z)} = r \cos^{2} θ + r \sin^{2} θ \frac{\partial (x, y, z)}{\partial (r, θ, z)} = r (\cos^{2} θ + \sin^{2} θ) \frac{\partial (x, y, z)}{\partial (r, θ, z)} = r \frac{\partial (x, y, z)}{\partial (r, θ, z)} = R . H . S$

Hence, proved.

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The formulas for converting from cylindrical coordinates to rectangular coordinates are x = r cos θ, y = r sin θ, and z = z. Prove that the Jacobian $\frac{\partial (x, y, z)}{\partial (r, θ, z)} = r$ .

Short Answer

Step by step solution

fgjf

Given information

Proof

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The formulas for converting from cylindrical coordinates to rectangular coordinates are x = r cos θ, y = r sin θ, and z = z. Prove that the Jacobian ∂(x,y,z)∂(r,θ,z)=r.

Short Answer

Step by step solution

fgjf

Given information

Proof

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Most popular questions from this chapter

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Probability and Statistics

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Applied Mathematics

Study anywhere. Anytime. Across all devices.

The formulas for converting from cylindrical coordinates to rectangular coordinates are x = r cos θ, y = r sin θ, and z = z. Prove that the Jacobian $\frac{\partial (x, y, z)}{\partial (r, θ, z)} = r$ .