Chapter 6: Q18P (page 295)

As in Problem 17, find the following gradients in two ways and show that your answers are equivalent $\nabla x$ .

Short Answer

Expert verified

The solution to this problem is $\nabla x = \hat{i}$ .

Step by step solution

Given Information.

The given information is $\nabla x$ .

Definition of Scalar field.

In mathematics and physics, scalar field or scalar-valued function referred to a scalarvalue to everypointin aspace– possiblyphysical space. The scalar may either be a (dimensionless)mathematical numberor aphysical quantity.

Find the solution.

$Use the equation \nabla f = e_{r} \frac{\partial f}{\partial r} + e_{θ} \frac{1}{r} \frac{\partial f}{\partial θ} + e_{z} \frac{\partial f}{\partial z} \nabla x = \nabla r \cos (θ) = \frac{\partial r \cos (θ)}{\partial r} e^{r} + \frac{\partial r \cos (θ)}{\partial r} \frac{1}{r} e_{θ} \nabla x = \cos (θ) e_{r} - \sin (θ) e_{θ} Simplify further . \nabla x = \cos (θ) [\cos (θ) \hat{i} + \sin (θ) \hat{j}] - \sin (θ) [- \sin (θ) \hat{i} + \cos (θ) \hat{j}] = \hat{i} Hence, the solution to this problem is \nabla x = \hat{i} .$

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As in Problem 17, find the following gradients in two ways and show that your answers are equivalent $\nabla x$ .

Short Answer

Step by step solution

Given Information.

Definition of Scalar field.

Find the solution.

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As in Problem 17, find the following gradients in two ways and show that your answers are equivalent ∇x.

Short Answer

Step by step solution

Given Information.

Definition of Scalar field.

Find the solution.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Physical Quantities And Units

Work Energy and Power

Electrostatics

Famous Physicists

Wave Optics

Particle Model of Matter

Study anywhere. Anytime. Across all devices.

As in Problem 17, find the following gradients in two ways and show that your answers are equivalent $\nabla x$ .