DEFINITIONS OF IMPORTANT "PARTIAL DIFFERENTIAL OPERATORS" IN VECTOR CALCULUS ---------------------Warren D. Smith----2003-------------------------------- All derivatives in this note will be partial. We will discuss the following 7 differential operators: name order input output ---- ----- ----- ------ grad 1 scalar-valued fn of vec vec-valued fn of vec directional derivative 1 scalar-valued fn of vec scal-valued fn of vec div 1 vector-valued fn of vec scal-valued fn of vec curl 1 vector-valued fn of vec vec-valued fn of vec laplacian 2 scalar-valued fn of vec scal-valued fn of vec jacobian 1 vector-valued fn of vec 3x3 matrix of fns of vec hessian 2 scalar-valued fn of vec 3x3 matrix of fns of vec All discussion will be in 3-dimensions but the generalization to any number of dimensions should be obvious (except that curl is an inherently 3D notion and hence does not generalize). The GRADIENT vector of a scalar-valued function of a vector is dF dF dF grad F(x,y,z) = ( --, --, -- ) where all derivatives are partial. dx dy dz 2 Example: if F(x,y,z) = x + x y + x z + z + 1 then grad F = (2x+y+z, x, x+1). We here may think of "grad" as a (partial) differential "operator", d d d TRIANGLE = ( --, --, -- ) dx dy dz that "operates" on scalar-valued functions of vectors (x,y,z) to produce a vector-valued function of (x,y,z). The DIRECTIONAL DERIVATIVE of F in direction D may be thought of as dF dF dF (D . TRIANGLE) F(x,y,z) = D1*-- + D2*-- + D3*-- dx dy dz meaning the (other) differential operator d d d (D . TRIANGLE) = ( D1*-- + D2*-- + D3*-- ) dx dy dz is applied to the function F. "Operators" are machines that input functions and output functions. Sort of functions of functions. 2 Example: if F(x,y,z) = x + x y + x z + z + 1 and D = (4,5,7) then (D . TRIANGLE) F = 4*(2x+y+z) + 5*x + 7*(x+1). We could also do dF1 dF2 dF3 TRIANGLE . F = --- + --- + --- dx dy dz where F = (F1,F2,F3) now is a VECTOR-valued function of a vector (x,y,z). (Note the dot.) This is also called "div F", the "divergence" of a vector field F. Note, div F is a scalar. Div is another useful differential operator. But we could also do the cross product of the TRIANGLE operator with F = (F1,F2,F3), a vector-valued function of a vector (x,y,z): dF3 dF2 dF1 dF3 dF1 dF2 TRIANGLE X F = ( --- - --- , --- - --- , --- - --- ) dy dz dz dx dy dx (Note the cross.) This is also called "curl F", the "curl" of a vector field F. Note, this is a vector. Curl is another useful differential operator. 2 2 Example: if F(x,y,z) = (x , xz, y z) then curl F = (2 y z - x, 0, z). Another important operator is the LAPLACIAN (TRIANGLE . TRIANGLE) 2 2 2 d F d F d F 2 laplacian F(x,y,z) = --- + --- + --- = TRIANGLE F. 2 2 2 dx dy dz The Laplacian is a "second order" differential operator (meaning: second derivatives are involved) that inputs scalar-valued functions of vectors and outputs scalar-valued functions of vectors. 2 Example: if F(x,y,z) = x + x y + x z + z + 1 then Laplacian F = 2. Another important operator is the "Jacobian matrix". This is the 3x3 matrix of first-order partial derivatives of every component F1,F2,F3 of a vector-valued function of a vector (x,y,z) in every axial direction x,y,z. [dF1 dF1 dF1] [--- --- ---] [dx dy dz ] [ ] [dF2 dF2 dF2] Jacobian F = [--- --- ---] [dx dy dz ] [ ] [dF3 dF3 dF3] [--- --- ---] [dx dy dz ] 2 Example: if F(x,y,z) = (x , x y, x z) then [2x 0 0] [ ] Jacobian F = [ y x 0] [ ] [ z 0 x] Finally, there is the "Hessian matrix." Given a scalar-valued function of a vector F(x,y,z), its Hessian is [ 2 2 2 ] [d F d F d F ] [---- ---- ----] [dxdx dxdy dxdz] [ ] [ 2 2 2 ] [d F d F d F ] Hessian F = [---- ---- ----] [dydx dydy dydz] [ ] [ 2 2 2 ] [d F d F d F ] [---- ---- ----] [dzdx dzdy dzdz] 2 Example: if F(x,y,z) = x + x y + x z + z + 1 then [2 1 1] Hessian F = [1 0 0] [1 0 0]. Note, due to equality of mixed partials, Hessian matrices are always "symmetric", i.e. the a-acrss, d-down entry is the same as the d-across, a-down entry. Also note, the sum of the diagonal entries of the Hessian (its "trace" in matrix-lingo) is the laplacian. A FEW USES OF THESE OPERATORS AND A FEW OF THEIR INTER-RELATIONS ---------------------------------------------------------------- curl grad F = TRIANGLE X TRIANGLE F = 0 basically because A X A = 0 causes everything to cancel out. THEOREM "The curl of a gradient is the 0-vector." Similarly div curl F = 0 basically since A . (A X B) = 0 causes everything to cancel out. THEOREM "The divergence of a curl is the 0-vector." Amazing true fact is that these 2 facts also work in reverse: i.e.: THEOREM(in 3D) "a vector field is a gradient of something if and only if its curl is the 0-vector." THEOREM(in 3D) "a vector field is a curl of something if and only if its div is 0." grad F is the direction of steepest increase of F (and its length tells you the rate of increase of F going in that direction). grad F is the 0-vector at "flat spots" such as mins, maxes, and saddle points. The Hessian of F is "positive definite" where F is concave-U. The Jacobian is extremely useful in "chain rule" situations...: JacobianMatrix [U(F(x,y,z))] = JacobianMatrix[U at F] JacobianMatrix[F at x,y,z]