Chapter 10: Q16P (page 528)

Use equations (9.2), (9.8), and (9.11) to evaluate the following expressions.In cylindrical coordinates $\nabla . e_{r}, \nabla . e_{θ}, \nabla \times e_{r}, \nabla \times e_{θ} .$ .

Short Answer

Expert verified

The required values are mentioned below.

$\nabla \cdot {\hat{e}}_{r} = \frac{1}{r} \nabla \cdot {\hat{e}}_{θ} = 0 \nabla \cdot {\hat{e}}_{r} = 0 \nabla \cdot {\hat{e}}_{θ} = \frac{1}{r} {\hat{e}}_{ϕ}$

Step by step solution

Definition of cylindrical coordinates.

The coordinate system primarily utilized in three-dimensional systems is the cylindrical coordinates of the system. The cylindrical coordinate system is used to find the surface area in three-dimensional space.

Determining the value of ∇.er,∇.eθ,∇×er,∇×eθ

Find the value of $\nabla . e_{r}, \nabla . e_{θ} .$ .

role="math" localid="1659260592443" $\nabla \cdot {\hat{e}}_{r} = \frac{1}{r} \frac{\partial}{\partial r} (r \cdot 1) \nabla \cdot {\hat{e}}_{r} = \frac{1}{r} \nabla \cdot {\hat{e}}_{θ} = \frac{1}{r} \frac{\partial}{\partial r} (1) \nabla \cdot {\hat{e}}_{θ} = 0$

Findthe value of $\nabla \times e_{r}, \nabla \times e_{θ} .$ .

$\nabla \cdot {\hat{e}}_{r} = \frac{\partial}{\partial r} (1) {\hat{e}}_{θ} - \frac{1}{r} \frac{\partial}{\partial r} (1) {\hat{e}}_{z} \nabla \cdot {\hat{e}}_{r} = 0 \nabla \cdot {\hat{e}}_{θ} = \frac{1}{r θ \sin} \frac{\partial}{\partial r} (r \cdot 1) {\hat{e}}_{r} + \frac{1}{r} \frac{\partial}{\partial r} (r \cdot 1) {\hat{e}}_{ϕ} \nabla \cdot {\hat{e}}_{θ} = \frac{1}{r} {\hat{e}}_{ϕ}$

The required values are mentioned below.

$\nabla \cdot {\hat{e}}_{r} = \frac{1}{r} \nabla \cdot {\hat{e}}_{θ} = 0 \nabla \cdot {\hat{e}}_{r} = 0 \nabla \cdot {\hat{e}}_{θ} = \frac{1}{r} {\hat{e}}_{ϕ}$

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Use equations (9.2), (9.8), and (9.11) to evaluate the following expressions.In cylindrical coordinates $\nabla . e_{r}, \nabla . e_{θ}, \nabla \times e_{r}, \nabla \times e_{θ} .$ .

Short Answer

Step by step solution

Definition of cylindrical coordinates.

Determining the value of ∇.er,∇.eθ,∇×er,∇×eθ

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Use equations (9.2), (9.8), and (9.11) to evaluate the following expressions.In cylindrical coordinates∇.er,∇.eθ,∇×er,∇×eθ. .

Short Answer

Step by step solution

Definition of cylindrical coordinates.

Determining the value of ∇.er,∇.eθ,∇×er,∇×eθ

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Kinematics Physics

Quantum Physics

Wave Optics

Magnetism and Electromagnetic Induction

Magnetism

Oscillations

Study anywhere. Anytime. Across all devices.

Use equations (9.2), (9.8), and (9.11) to evaluate the following expressions.In cylindrical coordinates $\nabla . e_{r}, \nabla . e_{θ}, \nabla \times e_{r}, \nabla \times e_{θ} .$ .