Gradients And Curls

Gradients and Curls

 

In the following, a and b are scalars, f, g, M, N are scalar functions of x, y, z,

G, F, F₁, F₂ are vector fields,
G = M x + N y, and

x, y, z are unit vectors along the cartesian coordinate x, y, z axes.

 

Term	Value
∇	(∂/∂x)x + (∂/∂y)y + (∂/∂z)z
∇f	the gradient of f
∇f	grad f
∇f	(∂f/∂x)x + (∂f/∂y)y + (∂f/∂z)z
curl grad f	∇ × ∇f
curl grad f	0 for f(x,y,z) with continuous 2nd partial derivatives
div F	the divergence of F
div F	∇ • F
div G	(∂M/∂x) + (∂N/∂y)
curl F	the curl of F
curl F	∇ × F
(curl G) • z	(∂N/∂x) - (∂M/∂y)
∇ (f g)	f ∇g + g ∇f
∇ • (g F)	g ∇ • F + ∇ g • F
∇ × (g F)	g ∇ × F + ∇ g × F
∇ • (a F1 + b F2)	a ∇ • F1 + b ∇ • F2
∇ × (a F1 + b F2)	a ∇ × F1 + b ∇ × F2
∇ (F1 • F2)	(F1 • ∇) F2    + (F2 • ∇) F1 + F1 × (∇ × F2) + F2 × (∇ × F1)
∇ (F1 × F2)	F2 • ∇ × F1 - F1 • ∇ × F2
∇ × (F1 × F2)	(F2 • ∇) F1 - (F1 • ∇) F2 + (∇ • F2) F1 - (∇ • F1) F2
∇ × (∇ × F)	∇ (∇ • F) - (∇ • ∇) F = ∇ (∇ • F) - ∇2 F
(∇ × F) × F	(F • ∇) F - (1/2) ∇ (F • F)
start learning	start learning

 

Reference: "University Calculus" by Hass, Weir, and Thomas, 2007 (ISBN=0-321-35014-6).