Chapter 13: Q. 52 (page 1080)

The formulas for converting from spherical coordinates to rectangular coordinates are $x = p \sin ϕ \cos θ, y = p \sin ϕ \sin θ, z = p \cos ϕ$ . Prove that the Jacobianrole="math" $\frac{\partial (x, y, z)}{\partial (p, θ, ϕ)} = - p^{2} \sin ϕ$ .

Short Answer

Expert verified

It is proven that $\frac{\partial (x, y, z)}{\partial (p, θ, ϕ)} = - p^{2} \sin ϕ$ .

Step by step solution

Given information

The transformation equations are,

$x = p \sin ϕ \cos θ, y = p \sin ϕ \sin θ, z = p \cos ϕ$

Proof

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

L . H . S = \frac{\partial (x, y, z)}{\partial (p, θ, ϕ)} \frac{\partial (x, y, z)}{\partial (p, θ, ϕ)} = |\begin{matrix} \frac{\partial x}{\partial p} & \frac{\partial y}{\partial p} & \frac{\partial z}{\partial p} \\ \frac{\partial x}{\partial θ} & \frac{\partial y}{\partial θ} & \frac{\partial z}{\partial θ} \\ \frac{\partial x}{\partial ϕ} & \frac{\partial y}{\partial ϕ} & \frac{\partial z}{\partial ϕ} \end{matrix}| \frac{\partial (x, y, z)}{\partial (p, θ, ϕ)} = |\begin{matrix} \sin ϕ \cos θ & \sin ϕ \sin θ & \cos ϕ \\ - p \sin ϕ \sin θ & p \sin ϕ \cos θ & 0 \\ p \cos ϕ \cos θ & p \cos ϕ \sin θ & - p \sin ϕ \end{matrix}| \frac{\partial (x, y, z)}{\partial (p, θ, ϕ)} = - p^{2} \sin^{3} ϕ - p^{2} \sin ϕ \cos^{2} ϕ \frac{\partial (x, y, z)}{\partial (p, θ, ϕ)} = - p^{2} \sin ϕ (\sin^{2} ϕ + \cos^{2} ϕ) \frac{\partial (x, y, z)}{\partial (p, θ, ϕ)} = - p^{2} \sin ϕ (1) \frac{\partial (x, y, z)}{\partial (p, θ, ϕ)} = - p^{2} \sin ϕ = R . H . S

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The formulas for converting from spherical coordinates to rectangular coordinates are $x = p \sin ϕ \cos θ, y = p \sin ϕ \sin θ, z = p \cos ϕ$ . Prove that the Jacobianrole="math" $\frac{\partial (x, y, z)}{\partial (p, θ, ϕ)} = - p^{2} \sin ϕ$ .

Short Answer

Step by step solution

Given information

Proof

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The formulas for converting from spherical coordinates to rectangular coordinates are x=psinϕcosθ,y=psinϕsinθ,z=pcosϕ. Prove that the Jacobianrole="math" ∂(x,y,z)∂(p,θ,ϕ)=-p2sinϕ.

Short Answer

Step by step solution

Given information

Proof

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Most popular questions from this chapter

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Calculus

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Study anywhere. Anytime. Across all devices.

The formulas for converting from spherical coordinates to rectangular coordinates are $x = p \sin ϕ \cos θ, y = p \sin ϕ \sin θ, z = p \cos ϕ$ . Prove that the Jacobianrole="math" $\frac{\partial (x, y, z)}{\partial (p, θ, ϕ)} = - p^{2} \sin ϕ$ .