If ϕ=3xy+2x2yz−12xz3ϕ=3xy+2x2yz−12xz3, Find ∇×(∇ϕ)
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
Find the divergence at (4,0,0)(4,0,0) for the vector field F(x,y,z)=exsinyi−excosyj+z2kF(x,y,z)=exsinyi−excosyj+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(sinzdx−cosxdy+sinydz) 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−sinx2z)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.
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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