V15.1 Del Operator 1. Symbolic notation: the del operator To have a compact notation, wide use is made of the symbolic operator “del” (some call it “nabla”): (1) ∇ = ∂ ∂ ∂ i+ j+ k ∂x ∂y ∂z ∂M ∂ Recall that the “product” of and the function M (x, y, z) is understood to be . Then ∂x ∂x we have ∂f ∂f ∂f i+ j+ k (2) grad f = ∇f = ∂x ∂y ∂z The divergence is sort of a symbolic scalar product: if F = M i + N j + P k , (3) div F = ∇ · F = ∂M ∂N ∂P + + ∂x ∂y ∂z while the curl, as we have noted, as a symbolic cross-product: � � � i j k �� ∂ ∂ ∂ � curl F = ∇ × F = �� ∂x ∂y ∂z � . �M N P � Notice how this notation reminds you that ∇ · F is a scalar function, while ∇ × F is a vector function. We may also speak of the Laplace operator (also called the “Laplacian”), deﬁned by lap f = ∇2 f = (∇ · ∇) f = (5) ∂2f ∂2f ∂2f + 2 + 2 . 2 ∂x ∂y ∂z Thus, Laplace’s equation may be written: ∇2 f = 0. (This is for example the equation satisﬁed by the potential function for an electrostatic ﬁeld, in any region of space where there are no charges; or for a gravitational ﬁeld, in a region of space where there are no masses.) In this notation, the divergence theorem and Stokes’ theorem are respectively �� (6) F · dS = ∇ · F dV F · dr = ∇ × F · dS S D C S Two important relations involving the symbolic operator are: (7) curl (grad f ) = 0 div curl F = 0 ′ ∇ × ∇f = 0 ∇·∇×F = 0 (7 ) The ﬁrst we have proved (it was part of the criterion for gradient ﬁelds); the second is an easy exercise. Note however how the symbolic notation suggests the answer, since we know that for any vector A, we have A × A = 0, A·A×F = 0 , and (7′ ) says this is true for the symbolic vector ∇ as well.