Scalar Potentials

The following are examples of scalar potential with the standard basis variables.


$\blacktriangleright$ Vector Calculus + Scalar Potential

(x, y, z), Scalar potential is ${\frac{{1}}{{2}}}$z2 + ${\frac{{1}}{{2}}}$y2 + ${\frac{{1}}{{2}% }}$x2

(x, z, y), Scalar potential is yz + ${\frac{{1}}{{2}}}$x2

(y, z, x), Scalar potential does not exist


In the next example, choose Evaluate, and then from the Vector Calculus submenu choose Scalar Potential. Because the vector field is a gradient, it has the original function as a scalar potential.


$\blacktriangleright$ Evaluate, Vector Calculus + Scalar Potential

∇$\left(\vphantom{ xy^{2}+yz^{3}}\right.$xy2 + yz3$\left.\vphantom{ xy^{2}+yz^{3}}\right)$ = $\left(\vphantom{ y^{2},2xy+z^{3},3yz^{2}}\right.$y2, 2xy + z3, 3yz2$\left.\vphantom{ y^{2},2xy+z^{3},3yz^{2}}\right)$, Scalar potential is xy2 + yz3


Note    When the vector field is a gradient, it has the original function as a scalar potential.


You would normally expect the scalar potential of the vector field  $\left(\vphantom{ cv,cu+2vw,v^2}\right.$cv, cu + 2vw, v2$\left.\vphantom{ cv,cu+2vw,v^2}\right)$ to be ucv + v2w; that is, you expect c to be treated as a constant. When the number of variables differs from the number of components in the field vector, a dialog box asks for the field variables. In this case, you can enter UserInputu,v,w to get the expected result. The dialog box also appears when you ask for the scalar potential of a vector field that specifies fewer than three variables, such as $\left(\vphantom{ y,x,0}\right.$y, x, 0$\left.\vphantom{ y,x,0}\right)$. Enter UserInputx,y,z in the dialog box to get the expected result xy for the scalar potential of this vector field.