Chapter 6: Q19P (page 295)

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

Short Answer

Expert verified

$\nabla y = \hat{j}$

Step by step solution

Given Information

The given information is $\nabla y$ .

Definition of Scalar field

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

Find the gradients

Use the equation . $\nabla f = e_{r} \frac{\partial f}{\partial r} + e_{θ} \frac{1}{r} \frac{\partial f}{\partial θ} + e_{z} \frac{\partial f}{\partial z}$

$\begin{array}{rcl} \nabla y & = & \nabla r s i n (θ) \\ = & \frac{\partial r s i n (θ)}{\partial r} e_{r} + \frac{\partial r s i n (θ)}{\partial r} \frac{1}{r} e_{θ} \nabla y \\ = & s i n (θ) e_{r} + \cos (θ) e_{θ} \end{array}$

Simplify further.

$\begin{array}{rcl} \nabla y & = & s i n (θ) [\cos (θ) \hat{i} + s i n (θ) \hat{j}] + \cos (θ) [- s i n (θ) \hat{i} + \cos (θ) \hat{j}] \\ = & \hat{j} \end{array}$

Hence, the solution to this problem is . $\nabla y = \hat{j}$

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

Short Answer

Step by step solution

Given Information

Definition of Scalar field

Find the gradients

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

Short Answer

Step by step solution

Given Information

Definition of Scalar field

Find the gradients

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Rotational Dynamics

Classical Mechanics

Electrostatics

Atoms and Radioactivity

Translational Dynamics

Electricity and Magnetism

Study anywhere. Anytime. Across all devices.

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