Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 6: Q19P (page 295)

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

Short Answer

Expert verified

∇y=j^

Step by step solution

01

Given Information

The given information is ∇y.

02

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.

03

Find the gradients

Use the equation .∇f=er∂f∂r+eθ1r∂f∂θ+ez∂f∂z

∇y=∇rsinθ=∂rsinθ∂rer+∂rsinθ∂r1reθ∇y=sinθer+cosθeθ

Simplify further.

∇y=sinθcosθi^+sinθj^+cosθ-sinθi^+cosθj^=j^

Hence, the solution to this problem is .∇y=j^

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Use Green’s theorem (Section 9) to do Problem 8.2.

∫C(yex-1)dx+exdy,where Cis the semicircle through(0,-10),(10,0),and(0,10)(Compare Problem 4.)

Find the derivative of xy2+yz at (1,1,2) in the direction of the vector 2i-j+2k.

IfF=xi+yj,calculate ∬F⋅ndσover the part of the surface z=4-x2-y2that is above the (x, y) plane, by applying the divergence theorem to the volume bounded by the surface and the piece that it cuts out of the (x,y)plane. Hint: What is F·non the (x,y)plane?

Suppose the density ρ varies from point to point as well as with time, that is, ρ=(x,y,z,t). If we follow the fluid along a streamline, then x,y,z are function of such that the fluid velocity is

v=idxdt+jdydt+kdzdt

Show that thendρ/dt=∂ρ/∂t+v−∇ρ . Combine this equation with (10.9)to get

ρ∇−v+dρdt=0

(Physically, is the rate of change of density with time as we follow the fluid along a streamline; ∂p/∂tis the corresponding rate at a fixed point.) For a steady state (that is, time-independent), ∂p/∂t=0, but ∂p/∂t is not necessarily zero. For an incompressible fluid, ∂p/∂t=0. Show that then role="math" localid="1657336080397" ∇×v=0. (Note that incompressible does not necessarily mean constant density since ∂p/∂t=0does not imply either time or space independence of ρ; consider, for example, a flow of watermixed with blobs of oil.)
See all solutions

Recommended explanations on Physics Textbooks

Dynamics

Read Explanation

Work Energy and Power

Read Explanation

Particle Model of Matter

Read Explanation

Conservation of Energy and Momentum

Read Explanation

Circular Motion and Gravitation

Read Explanation

Electricity

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free