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Magazine Problem 108
For the following exercises, determine whether the vector field is conservative and, if it is, find the potential function. $$ \mathbf{F}(x, y)=\left(e^{2 x} \sin y\right) \mathbf{i}+\left[e^{2 x} \cos y\right] \mathbf{j} $$
Problem 109
For the following exercises, determine whether the vector field is conservative and, if it is, find the potential function. $$ \mathbf{F}(x, y)=(6 x+5 y) \mathbf{i}+(5 x+4 y) \mathbf{j} $$
Problem 111
For the following exercises, determine whether the vector field is conservative and, if it is, find the potential function. $$ \mathbf{F}(x, y)=\left[y e^{x}+\sin (y)\right] \mathbf{i}+\left[e^{x}+x \cos (y)\right] \mathbf{j} $$
Problem 116
For the following exercises, determine whether the vector field is conservative and, if so, find a potential function. $$ \mathbf{F}(x, y)=(12 x y) \mathbf{i}+6\left(x^{2}+y^{2}\right) \mathbf{j} $$
Problem 117
For the following exercises, determine whether the vector field is conservative and, if so, find a potential function. $$ \mathbf{F}(x, y)=\left(e^{x} \cos y\right) \mathbf{i}+6\left(e^{x} \sin y\right) \mathbf{j} $$
Problem 118
For the following exercises, determine whether the vector field is conservative and, if so, find a potential function. $$ \mathbf{F}(x, y)=\left(2 x y e^{x^{2} y}\right) \mathbf{i}+6\left(x^{2} e^{x^{2} y}\right) \mathbf{j} $$
Problem 119
For the following exercises, determine whether the vector field is conservative and, if so, find a potential function. $$ \mathbf{F}(x, y, z)=\left(y e^{z}\right) \mathbf{i}+\left(x e^{z}\right) \mathbf{j}+\left(x y e^{z}\right) \mathbf{k} $$
Problem 121
For the following exercises, determine whether the vector field is conservative and, if so, find a potential function. $$ \mathbf{F}(x, y, z)=\left(\frac{1}{y}\right) \mathbf{i}+\left(\frac{x}{y^{2}}\right) \mathbf{j}+(2 z-1) \mathbf{k} $$
Problem 122
For the following exercises, determine whether the vector field is conservative and, if so, find a potential function. $$ \mathbf{F}(x, y, z)=3 z^{2} \mathbf{i}-\cos y \mathbf{j}+2 x z \mathbf{k} $$
Problem 123
For the following exercises, determine whether the vector field is conservative and, if so, find a potential function. $$ \mathbf{F}(x, y, z)=(2 x y) \mathbf{i}+\left(x^{2}+2 y z\right) \mathbf{j}+y^{2} \mathbf{k} $$
