Question

\({\rm{F(x,y,z) = sinyi + xcosyj - sinzk}}{\rm{.}}\)

Short answer

\({\rm{F}}\) is a conservative vector field The potential function is \({\rm{f(x,y,z) = xsiny + cosz + K}}{\rm{.}}\)

Step-by-step solution

Step 1:\({\rm{F}}\) is conservative,

The vector field \({\rm{F}}\)\({{\rm{R}}^{\rm{3}}}\) is conservative if and only if the curl \({\rm{F = 0}}\)

By definition curl

\({\rm{F = }}\nabla {\rm{ \times F}}\)

\({\rm{F = Pi + Qj + Rk}}\)

Then \({\rm{F}}\) is conservative if and only if

\(\left( {\begin{array}{*{20}{c}}{\rm{i}}&{\rm{j}}&{\rm{k}}\\{\frac{\partial }{{\partial {\rm{x}}}}}&{\frac{\partial }{{\partial {\rm{y}}}}}&{\frac{\partial }{{\partial {\rm{z}}}}}\\{\rm{P}}&{\rm{Q}}&{\rm{R}}\end{array}} \right) = 0\)

Which is

\(\frac{{\partial {\rm{R}}}}{{\partial {\rm{y}}}}{\rm{ - }}\frac{{\partial {\rm{Q}}}}{{\partial {\rm{z}}}}\;\;\;{\rm{i - }}\frac{{\partial {\rm{P}}}}{{\partial {\rm{z}}}}{\rm{ - }}\frac{{\partial {\rm{R}}}}{{\partial {\rm{x}}}}\;\;\;{\rm{j + }}\frac{{\partial {\rm{Q}}}}{{\partial {\rm{x}}}}{\rm{ - }}\frac{{\partial {\rm{P}}}}{{\partial {\rm{y}}}}\;\;\;{\rm{k = 0}}\)

This implies that \({\rm{F}}\) is conservative if and only if the following conditions are met

\(\frac{{\partial {\rm{R}}}}{{\partial {\rm{y}}}}{\rm{ = }}\frac{{\partial {\rm{Q}}}}{{\partial {\rm{z}}}}\)and \(\frac{{\partial {\rm{P}}}}{{\partial {\rm{z}}}}{\rm{ = }}\frac{{\partial {\rm{R}}}}{{\partial {\rm{x}}}}\;\;\;\)and \(\frac{{\partial {\rm{Q}}}}{{\partial {\rm{x}}}}{\rm{ = }}\frac{{\partial {\rm{P}}}}{{\partial {\rm{y}}}}\)

\({R_y} = {Q_z}\) and \({P_z} = {R_x}\) and \({Q_x} = {P_y}\)

The vector field is conservative,

Given that

\({\rm{F(x,y,z) = sinyi + xcosyj - sinzk}}\)

Then have

\({\rm{P = siny}} \Rightarrow {{\rm{P}}_{\rm{z}}}{\rm{ = 0 and }}{{\rm{P}}_{\rm{y}}}{\rm{ = cosy}}\)

\({\rm{Q = xcosy}} \Rightarrow {{\rm{Q}}_{\rm{z}}}{\rm{ = 0 and }}{{\rm{Q}}_{\rm{x}}}{\rm{ = cosy}}\)

\({\rm{R = - sinz}} \Rightarrow {{\rm{R}}_{\rm{x}}}{\rm{ = 0 and }}{{\rm{R}}_{\rm{y}}}{\rm{ = 0}}\)

Note that

Have \({{\rm{R}}_{\rm{y}}}{\rm{ = }}{{\rm{Q}}_{\rm{z}}}{\rm{ = 0 }}\) and \({{\rm{P}}_{\rm{z}}}{\rm{ = }}{{\rm{R}}_{\rm{x}}}{\rm{ = 0}}\)\({{\rm{Q}}_{\rm{x}}}{\rm{ = }}{{\rm{P}}_{\rm{y}}}{\rm{ = cosy}}\)

Therefore, the given vector field is conservative.

Equating,

\({\rm{F(x,y,z) = }}\nabla {\rm{f}}\)

\({{\rm{f}}_{\rm{x}}}{\rm{i + }}{{\rm{f}}_{\rm{y}}}{\rm{j + }}{{\rm{f}}_{\rm{z}}}{\rm{k = sinyi + xcosyj - sinzk}}\)

Therefore,

\({{\rm{f}}_{\rm{x}}}{\rm{ = siny}}\)

And

\({{\rm{f}}_{\rm{y}}}{\rm{ = xcosy}}\)

And

\({{\rm{f}}_{\rm{z}}}{\rm{ = - sinz}}\)

Integrate concerning\({\rm{x,}}\)

\({{\rm{f}}_{\rm{x}}}{\rm{ = siny}}\)

It integratesconcerns\({\rm{x}}\) while treating \({\rm{y}}\) and \({\rm{z}}\) a constant

\({\rm{f(x,y) = xsiny + p(y,z)}} \to {\rm{(1)}}\)

Note that \({\rm{p(y, z)}}\)this is the Integration constant.

Differentiating this equation concerning \({\rm{z,}}\)          

Partial differentiate Equation \({\rm{(1)}}\)concerning\(y\) to get

\({{\rm{f}}_{\rm{y}}}{\rm{ = xcosy + }}{{\rm{p}}_{\rm{y}}}{\rm{(y,z)}}\)

But we know that \({{\rm{f}}_{\rm{y}}}{\rm{ = xcosy}}\)

Therefore,

\({{\rm{p}}_{\rm{y}}}{\rm{ = 0}}\)

Integrate concerning\({\rm{y}}\) assuming \({\rm{z}}\) is constant, to get

\({\rm{p(y,z) = g(z)}}\)

Differentiating this equation concerning to \({\rm{z,}}\)gives

\({{\rm{p}}_{\rm{z}}}{\rm{ = }}{{\rm{g}}^{'}}{\rm{(z)}}\)

Partial differentiate Equation\({\rm{(1)}}\) concerning \({\rm{z,}}\)

Partial differentiate Equation \({\rm{(1)}}\) concerning \({\rm{z,}}\) getting

\({{\rm{f}}_{\rm{z}}}{\rm{ = }}{{\rm{p}}_{\rm{z}}}\)

But we know that \({{\rm{f}}_{\rm{z}}}{\rm{ = - sinz}}\)

Therefore,

\({{\rm{p}}_{\rm{z}}}{\rm{(y,z) = - sinz}}\)

But know that \({{\rm{p}}_{\rm{z}}}{\rm{ = }}{{\rm{g}}^{'}}{\rm{(z)}}\)

Therefore, have

\({{\rm{g}}^{'}}{\rm{(z) = - sinz}}\)

Integrate, to get \({\rm{g(z) = cosz + K}}\)

Therefore, \({\rm{p(y,z) = cosz + K}}\)

Substitute \({\rm{p(y,z)}}\)in Equation \({\rm{(1)}}\),

\({\rm{f(x,y,z) = xsiny + cosz + K}}\)

\({\rm{F}}\) is a conservative vector field The potential function is \({\rm{f(x,y,z) = xsiny + cosz + K}}{\rm{.}}\)