91Ó°ÊÓ

In Exercises 1 through 20 , evaluate the line integral over the given curve. \(\int_{c} z d x+x d y+y d z ; C:\) the circular helix \(\mathbf{R}(t)=a \cos t^{\circ}+a \sin t^{\circ}+t k ; 0 \leq t \leq 2 \pi\)

Find three numbers whose sum is \(N(N>0)\) such that their product is as great as possible.

In Exercises 13 through 18 , if the two given surfaces intersect in a curve, find equations of the tangent line to the curve of intersection at the given point; if the two given surfaces are tangent at the given point, prove it. \(x^{2}+y^{2}+z^{2}=8, y z=4 ;(0,2,2)\)

\(f(x, y)=e^{x} \cos y+e^{y} \sin x ; P(1,0), Q(-3,3)\)

In Exercises 1 through 20 , evaluate the line integral over the given curve. \(\int_{c} 2 x y d x+\left(6 y^{2}-x z\right) d y+10 z d z ; C:\) the twisted cubic \(R(t)=t i+t^{2} j+t^{3} k, 0 \leq t \leq 1\)

Prove that the box having the largest volume that can be placed inside a sphere is in the shape of a cube.

Prove that every normal line to the sphere \(x^{2}+y^{2}+z^{2}=a^{2}\) passes through the center of the sphere.

\(\int_{C}(2 x \cos y-3) d x-\left(x^{2} \sin y+z^{2}\right) d y-(2 y z-2) d z ; A\) is \((-1,0,3)\) and \(B\) is \((1, \pi, 0) ;\) Exercise 9

In the following exercises, determine if the vector is a gradient. If it is, find a function having the given gradient \(z \tan y \mathbf{i}+x z \sec ^{2} y \mathbf{j}+x \tan y \mathbf{k}\)

\(f(x, y, z)=x-2 y+z^{2} ; P(3,1,-2), Q(10,7,4)\)