Answered: If ϕ=3xy+2x2yz−12xz3ϕ=3xy+2x2yz−12xz3,…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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If ϕ=3xy+2x2yz−12xz3ϕ=3xy+2x2yz−12xz3, Find ∇×(∇ϕ)

The vector field F(x,y)=xi+yjx2+y2F(x,y)=xi+yjx2+y2 is solenoidal what points?

Find the divergence at (4,0,0)(4,0,0) for the vector field F(x,y,z)=exsinyi−excosyj+z2kF(x,y,z)=exsin⁡yi−excos⁡yj+z2k

Consider the conservative vector field F(x,y,z)=(yz+2x)i+(xz+2y)j+(xy+2z)k.F(x,y,z)=(yz+2x)i+(xz+2y)j+(xy+2z)k. Evaluate ∫cF⋅dr,∫cF⋅dr, c: is a curve starting at (0,0,0)(0,0,0) to (1,1,1).(1,1,1).

Use Stoke's theorem to evaluate ∮c(sinzdx−cosxdy+sinydz)∮c(sin⁡zdx−cos⁡xdy+sin⁡ydz) where c is the boundary of the rectangule 0≤x≤π,0≤y≤1,z=3.

Determine whether the vector field is conservative. If it is, find a potential function for the vector field.

F(x,y,z)=7x6y8z9i+8x7y7z9j+9x7y8z8kF(x,y,z)=7x6y8z9i+8x7y7z9j+9x7y8z8k

Given a vector field F(x,y,z)=(3x+z)i+(y2−sinx2z)j+(xz+yex5)k,F(x,y,z)=(3x+z)i+(y2−sin⁡x2z)j+(xz+yex5)k, the divengence theorem value in the region 0<x<1,0<y<3,and0<z<20<x<1,0<y<3,and0<z<2 is what?

Suppose r=xi+yj+zkr=xi+yj+zk and r=|r|,r=|r|, then div(gradr3)div(gradr3) is what?

Find the work done by the vector field F(x,y)=y3i+(x3+3xy2)jF(x,y)=y3i+(x3+3xy2)j using Green's Theorem, in moving a particle once in an anticlockwise direction around the circle of radius "Q" centred at the origin.

Evaluate the line integral along the given path. ∫c(4x−5y)dsc:r(t)=3ti+2tj,0≤y≤6

Find the total mass of the wire with density \(\rho.

r(t)=t2i+2tj,ρ(x,y)=11y12,0≤t≤1r(t)=t2i+2tj,ρ(x,y)=11y12,0≤t≤1

Suppose FF is a vector field, then ∇×(∇×F)∇×(∇×F) is equal to

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