By steady state temperature distributions t f x y z

This preview shows page 7 - 10 out of 13 pages.

by steady-state temperature distributionsT=f(x, y, z) in space, by grav-itational potentials, and by electrostatic potentials.The two-dimensionalLaplace equation2f∂x2+2f∂y2= 0 describes potentials and steady-state tem-perature distributions in a plane. The plane may be treated as a thin sliceof the solid perpendicular to thez-axis.Show that each of the followingfunctions satisfies a Laplace equation.(a)f(x, y) = lnx2+y2(b)f(x, y, z) =e3x+4ycos 5z39. For the following exercises,(i) expressdω/dtas a function oftboth by using the chain rule and byexpressingωin terms oftand differentiating directly with respect tot.(ii) Evaluatedtat the given value oft.40. Letz= tan-1(x/y),x=ucosv,y=usinv. Express∂z/∂uand∂z/∂vas functions ofuandvboth by using the Chain Rule and by expressingzdirectly in terms ofuandvbefore differentiating. Then evaluate∂z/∂uand∂z/∂vat the point (u, v) = (1.3, π/6).41. Letω= ln(x2+y2+z2),x=uevsinu,y=uevcosu,z=uev. Express∂ω/∂uand∂ω/∂vas functions ofuandvboth by using the Chain Ruleand by expressingωdirectly in terms ofuandvbefore differentiating. Thenevaluate∂ω/∂uand∂ω/∂vat the point (u, v) = (-2,0).7
Background image
42. Letu=eqrsin-1p,p= sinx,q=z2lny,r= 1/z. Express∂u/∂x,∂u/∂yand∂u/∂zas functions ofx yandzboth by using the Chain Rule and byexpressingudirectly in terms ofx,yandzbefore differentiating.Thenevaluate∂u/∂x,∂u/∂yand∂u/∂zat the point (x, y, z) = (π/4,1/2,-1/2).43. For the following exercises, draw a tree diagram and write a Chain Ruleformula for each derivative.(a)dzdtforz=f(u, v, ω), u=g(t), v=h(t), ω=k(t)(b)∂ω∂xand∂ω∂yforω=f(r, s, t), r=g(x, y), s=h(x, y), t=k(x, y)(c)∂ω∂xand∂ω∂yforω=g(u, v), u=h(x, y), v=k(x, y)(d)∂y∂rfory=f(u), u=g(r, s)(e)∂ω∂pforω=f(x, y, z, v), x=g(p, q), y=h(p, q), z=j(p, q), v=k(p, q)(f)∂ω∂sforω=g(x, y), x=h(r, s, t), y=k(r, s, t)44. Assuming that the following equations defineyas a differentiable functionofx, use the formula for implicit differentiation to find the value ofdy/dxatthe given point.45. Assuming that the following equations definezas a differentiable function ofxandy, use the formula for implicit differentiation to find the value of∂z∂xand∂z∂yat the given point.8
Background image
46. Find∂z∂uwhenu= 0,v= 1 ifz= sinxy+xsiny,x=u2+v2,y=uv.47. Find∂z∂uand∂z∂vwhenu= 1,v=-2 ifz= lnqandq=v+ 3 tan-1u.48. Iff(u, v, ω) is differentiable andu=x-y,v=y-z, andω=z-x, showthat∂f∂x+∂f∂y+∂f∂z= 0.49. Find the gradient of the function at the given point. Then sketch the gradienttogether with the level curve that passes through the point.
Background image
Image of page 10

You've reached the end of your free preview.

Want to read all 13 pages?

  • Fall '19

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern

Stuck? We have tutors online 24/7 who can help you get unstuck.
A+ icon
Ask Expert Tutors You can ask You can ask You can ask (will expire )
Answers in as fast as 15 minutes