# By steady state temperature distributions t f x y z

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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
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
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.

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