"Vector calculus identities"

grad(v) = d(v,(x,y,z))

div(v) = contract(grad(v))

cross(u,v) = (
  u[2] v[3] - u[3] v[2],
  u[3] v[1] - u[1] v[3],
  u[1] v[2] - u[2] v[1]
)

curl(f) = (
  d(f[3],y) - d(f[2],z),
  d(f[1],z) - d(f[3],x),
  d(f[2],x) - d(f[1],y)
)

laplacian(f) = d(f,x,x) + d(f,y,y) + d(f,z,z)

F = (FX(x,y,z),FY(x,y,z),FZ(x,y,z))
G = (GX(x,y,z),GY(x,y,z),GZ(x,y,z))

"1. Verify div curl F = 0"
check(div(curl(F)) == 0)

"2. Verify curl grad f = 0"
check(curl(grad(f(x,y,z))) == 0)

"3. Verify div grad f = laplacian f"
check(div(grad(f(x,y,z))) == laplacian(f(x,y,z)))

"4. Verify curl curl F = grad div F - laplacian F"
check(curl(curl(F)) == grad(div(F)) - laplacian(F))

"5. Verify grad(f g) = f grad g + g grad f"
check(grad(f(x,y,z) g(x,y,z)) == f(x,y,z) grad(g(x,y,z)) + g(x,y,z) grad(f(x,y,z)))

"6. Verify grad(F . G) = (grad F) . G + (grad G) . F + F x curl G + G x curl F"
check(grad(dot(F,G)) == dot(grad(F),G) + dot(grad(G),F) + cross(F,curl(G)) + cross(G,curl(F)))

"7. Verify div(f F) = f div F + F . grad f"
check(div(f(x,y,z) F) == f(x,y,z) div(F) + dot(F,grad(f(x,y,z))))

"8. Verify div(F x G) = G . curl F - F . curl G"
check(div(cross(F,G)) == dot(G,curl(F)) - dot(F,curl(G)))

"9. Verify curl(f F) = f curl F - F x grad f"
check(curl(f(x,y,z) F) == f(x,y,z) curl(F) - cross(F,grad(f(x,y,z))))

"10. Verify curl(F x G) = F div G - G div F + (grad F) . G - (grad G) . F"
check(curl(cross(F,G)) == F div(G) - G div(F) + dot(grad(F),G) - dot(grad(G),F))

"ok"