Step 1

To Determine:

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

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

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

\(\displaystyle{d}{f}={y}{z}{\left.{d}{x}\right.}+{x}{z}{\left.{d}{y}\right.}+{x}{y}{\left.{d}{z}\right.}\) for some f, and the integral value is f(3,5,0) -f(1,1,2)

Step 2

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

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

\(\displaystyle={3}{\left[{\left({3}\times{5}\times{0}\right)}-{\left({1}\times{1}\times{2}\right)}\right]}=-{6}\)

To Determine:

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

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

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

\(\displaystyle{d}{f}={y}{z}{\left.{d}{x}\right.}+{x}{z}{\left.{d}{y}\right.}+{x}{y}{\left.{d}{z}\right.}\) for some f, and the integral value is f(3,5,0) -f(1,1,2)

Step 2

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

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

\(\displaystyle={3}{\left[{\left({3}\times{5}\times{0}\right)}-{\left({1}\times{1}\times{2}\right)}\right]}=-{6}\)