91Ó°ÊÓ

In Exercises 1 through 20 , evaluate the line integral over the given curve. \(\int_{C} y d x+x d y ; C: \mathbf{R}(t)=t \mathbf{i}+t^{2} \mathbf{j}, 0 \leq t \leq 1\)

In Exercises 1 through 10 , prove that the given force field is conservative and find a potential function. \(\mathbf{F}(x, y, z)=\left(x^{2}-y\right) \mathbf{i}-(x-3 z) \mathbf{j}+(z+3 y) \mathbf{k}\)

In the following exercises, determine if the vector is a gradient. If it is, find a function having the given gradient \(\left(2 x y+y^{2}+1\right) \mathbf{i}+\left(x^{2}+2 x y+x\right) \mathbf{j}\)

In Exercises 1 through 12 , find an equation of the tangent plane and equations of the normal line to the given surface at the indicated point. \(z=x^{1 / 2}+y^{1 / 2} ;(1,1,2)\)

In Exercises 1 through 10 , prove that the given force field is conservative and find a potential function. \(\mathbf{F}(x, y, z)=\left(2 y^{3}-8 x z^{2}\right) \mathbf{i}+\left(6 x y^{2}+1\right) \mathbf{j}-\left(8 x^{2} z+3 z^{2}\right) \mathbf{k}\)

In Exercises 7 through 12 , use the method of Lagrange multipliers to find the critical points of the given function subject to the indicated constraint. \(f(x, y)=x^{2}+x y+2 y^{2}-2 x\) with constraint \(x-2 y+1=0\)

In the following exercises, determine if the vector is a gradient. If it is, find a function having the given gradient \(\left(\frac{1}{x^{2}}+\frac{1}{y^{2}}\right) \mathrm{i}+\left(\frac{1-2 x}{y^{3}}\right) \mathrm{j}\)

In Exercises 1 through 12 , find an equation of the tangent plane and equations of the normal line to the given surface at the indicated point. \(x^{1 / 2}+y^{1 / 2}+z^{1 / 2}=4 ;(4,1,1)\)

In Exercises 1 through 20 , evaluate the line integral over the given curve. \(\int_{C}(x-y) d x+(y+x) d y ; C:\) the entire circle \(x^{2}+y^{2}=4\)

In Exercises 1 through 10 , prove that the given force field is conservative and find a potential function. \(\mathbf{F}(x, y, z)=(2 x \cos y-3) \mathbf{i}-\left(x^{2} \sin y+z^{2}\right) \mathbf{j}-(2 y z-2) \mathbf{k}\)