Q. 51 The formulas for converting from... [FREE SOLUTION]

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Chapter 13: Q. 51 (page 1080)

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

Short Answer

Expert verified

It is proven that $\frac{鈭� (x, y, z)}{鈭� (r, 胃, z)} = r$

Step by step solution

fgjf

gfh

Given information

The transformation equations are,

$x = r \cos 胃, y = r \sin 胃, z = z$

Proof

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

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

Hence, proved.

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91影视

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

Short Answer

Step by step solution

fgjf

Given information

Proof

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91影视

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

Short Answer

Step by step solution

fgjf

Given information

Proof

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Pure Maths

Discrete Mathematics

Theoretical and Mathematical Physics

Statistics

Calculus

Study anywhere. Anytime. Across all devices.

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