Chapter 10: Q19P (page 528)

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

Short Answer

Expert verified

The required values are mentioned below.

$\nabla \times (r {\hat{e}}_{θ}) = 2 r {\hat{e}}_{ϕ} \nabla (r \cos θ) = \cos θ {\hat{e}}_{r} - \sin θ {\hat{e}}_{ϕ} \nabla \cdot r = 3$

Step by step solution

Definition of spherical coordinates.

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

Determining the value of ∇×(reθ),∇(rcosθ),∇.r

Findthe value of $\nabla \times (r {\hat{e}}_{θ})$ .

role="math" localid="1659261996986" $\nabla \times (r {\hat{e}}_{θ}) = - \frac{1}{r^{2} \sin θ \partial ϕ} (r^{2}) {\hat{e}}_{r} + \frac{1}{r} \frac{\partial}{\partial r} r^{2} {\hat{e}}_{ϕ} \nabla \times (r {\hat{e}}_{θ}) = 2 r {\hat{e}}_{ϕ}$

Findthe value of .

role="math" localid="1659262028625" $\nabla (r \cos θ) = \frac{\partial}{\partial r} (r \cos θ) {\hat{e}}_{r} + \frac{1}{r} \frac{\partial}{\partial θ} (r \cos θ) {\hat{e}}_{r} \nabla (r \cos θ) = \cos θ {\hat{e}}_{r} - \sin θ {\hat{e}}_{r}$

Findthe value of .

$\nabla \cdot r = \frac{1}{r^{2} s i n θ} \frac{\partial}{\partial r} (r^{2} s i n θ r) = 3$

The required values are mentioned below.

role="math" localid="1659262073399" $\nabla \times (r {\hat{e}}_{θ}) = 2 r {\hat{e}}_{ϕ} \nabla (r \cos θ) = \cos θ {\hat{e}}_{r} - \sin θ {\hat{e}}_{r} \nabla \cdot r = \frac{1}{r^{2} s i n θ} \frac{\partial}{\partial r} (r^{2} s i n θ r) = 3$

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

Short Answer

Step by step solution

Definition of spherical coordinates.

Determining the value of ∇×(reθ),∇(rcosθ),∇.r

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Use equations (9.2), (9.8), and (9.11) to evaluate the following expressions. In spherical coordinates ∇×(reθ),∇(rcosθ),∇.r.

Short Answer

Step by step solution

Definition of spherical coordinates.

Determining the value of ∇×(reθ),∇(rcosθ),∇.r

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Linear Momentum

Solid State Physics

Kinematics Physics

Particle Model of Matter

Electromagnetism

Magnetism

Study anywhere. Anytime. Across all devices.

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