Step 1

To Determine:

Show that the differential forms in the integrals are exact. Then evaluate the integrals.

Given: we have an integral \(\displaystyle{\int_{{{\left({1},{1},{2}\right)}}}^{{{\left({3},{5},{0}\right)}}}}{y}{z}{\left.{d}{x}\right.}+{x}{z}{\left.{d}{y}\right.}+{x}{y}{\left.{d}{z}\right.}\)

Explanation: let M=yz , N=xz ,P=xy and apply the Test for exactness

\(\displaystyle{\frac{{\partial{P}}}{{\partial{y}}}}={x}={\frac{{\partial{N}}}{{\partial{z}}}}\)

\(\displaystyle{\frac{{\partial{M}}}{{\partial{z}}}}={y}={\frac{{\partial{P}}}{{\partial{x}}}}\)

\(\displaystyle{\frac{{\partial{N}}}{{\partial{x}}}}={z}={\frac{{\partial{M}}}{{\partial{y}}}}\)

so this tells that the given differential form is exact. now let us consider that

df=yz dx+xz dy+xy dz for some f, and the integral value is f(3,5,0) -f(1,1,2)

Step 2

\(\displaystyle{\int_{{{\left({1},{1},{2}\right)}}}^{{{\left({3},{5},{0}\right)}}}}{y}{z}{\left.{d}{x}\right.}+{x}{z}{\left.{d}{y}\right.}+{x}{y}{\left.{d}{z}\right.}={{\left[{y}{z}{x}+{x}{z}{y}+{x}{y}{z}\right]}_{{{\left({1},{1},{2}\right)}}}^{{{\left({3},{5},{0}\right)}}}}\)

\(\displaystyle={{\left[{3}{x}{y}{z}\right]}_{{{\left({1},{1},{2}\right)}}}^{{{\left({3},{5},{0}\right)}}}}\)

\(\displaystyle={3}{\left[{\left({3}\times{5}\times{0}\right)}-{\left({1}\times{1}\times{2}\right)}\right]}\)

=-6

To Determine:

Show that the differential forms in the integrals are exact. Then evaluate the integrals.

Given: we have an integral \(\displaystyle{\int_{{{\left({1},{1},{2}\right)}}}^{{{\left({3},{5},{0}\right)}}}}{y}{z}{\left.{d}{x}\right.}+{x}{z}{\left.{d}{y}\right.}+{x}{y}{\left.{d}{z}\right.}\)

Explanation: let M=yz , N=xz ,P=xy and apply the Test for exactness

\(\displaystyle{\frac{{\partial{P}}}{{\partial{y}}}}={x}={\frac{{\partial{N}}}{{\partial{z}}}}\)

\(\displaystyle{\frac{{\partial{M}}}{{\partial{z}}}}={y}={\frac{{\partial{P}}}{{\partial{x}}}}\)

\(\displaystyle{\frac{{\partial{N}}}{{\partial{x}}}}={z}={\frac{{\partial{M}}}{{\partial{y}}}}\)

so this tells that the given differential form is exact. now let us consider that

df=yz dx+xz dy+xy dz for some f, and the integral value is f(3,5,0) -f(1,1,2)

Step 2

\(\displaystyle{\int_{{{\left({1},{1},{2}\right)}}}^{{{\left({3},{5},{0}\right)}}}}{y}{z}{\left.{d}{x}\right.}+{x}{z}{\left.{d}{y}\right.}+{x}{y}{\left.{d}{z}\right.}={{\left[{y}{z}{x}+{x}{z}{y}+{x}{y}{z}\right]}_{{{\left({1},{1},{2}\right)}}}^{{{\left({3},{5},{0}\right)}}}}\)

\(\displaystyle={{\left[{3}{x}{y}{z}\right]}_{{{\left({1},{1},{2}\right)}}}^{{{\left({3},{5},{0}\right)}}}}\)

\(\displaystyle={3}{\left[{\left({3}\times{5}\times{0}\right)}-{\left({1}\times{1}\times{2}\right)}\right]}\)

=-6