HP 48gII hp 48gII_user's manual_English_E_HDPMSG48E67_V2.pdf - Page 477

As an example, in an earlier example we attempted to find a potential function for the vector field F(x,y,z) = (x+y)i + (x-y+z)j + xzk, and got an error message back from function POTENTIAL. To verify that this is a rotational field (i.e., ∇×F ≠ 0), we use function CURL on this field: On the other hand, the vector field F(x,y,z) = xi + yj + zk, is indeed irrotational as shown below: Vector potential Given a vector field F(x,y,z) = f(x,y,z)i+g(x,y,z)j+h(x,y,z)k, if there exists a vector function Φ(x,y,z) = φ(x,y,z)i+ψ(x,y,z)j+η(x,y,z)k, such that F = curl then function Φ(x,y,z) is referred to as the vector potential of F(x,y,z). The calculator provides function VPOTENTIAL, available through the command catalog (,N), to calculate the vector potential, Φ(x,y,z), given the vector field, F(x,y,z) = f(x,y,z)i+g(x,y,z)j+h(x,y,z)k. For example, given the vector field, F(x,y,z) = -(yi+zj+xk), function VPOTENTIAL produces i.e., Φ(x,y,z) = -x2/2j + (-y2/2+zx)k. It should be indicated that there is more than one possible vector potential functions Φ for a given vector field F. For example, the following screen shot shows that the curl of the vector function Φ1 = [X2+Y2+Z2,XYZ,X+Y+Z] is the vector F = ∇× Φ2 = [1-XY,2Z-1,ZY-2Y]. Application of function VPOTENTIAL Page 15-6