Transcription of Gradient, divergence, and curl Math 131 Multivariate Calculus
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Gradient, divergence, and curl Math 131 Multivariate Calculus
Definition 2. We define the curl of a vector field in space, F : R3 R3 , as curl F = F .. Gradient, divergence, and curl = , , (F1 , F2 , F3 ). x y z Math 131 Multivariate Calculus i j k D Joyce, Spring 2014 . = x y z F1 F2 F3.. The del operator . First, we'll start by ab- F3 F2 F1 F3 F2 F1. = , , . stracting the gradient to an operator. By the y z z x x y way, the gradient of f isn't always denoted f ;. sometimes it's denoted grad f . We'll look at a couple of examples of curl in class, As you know the gradient of a scalar field f : too. It's harder to get a good intuition for curl, but n R R is it does say something about how much and which way a vector field swirls, or rotates. A vector field . f f f whose curl is constantly 0 is called irrotational. f = , ,.., . You can take curls of plane vector fields F : R2.
r= @ @x 1; @ @x 2;:::; @ @x n which, when applied to f yields rf. This ris called the del operator. We can treat this del operator like a vector itself. We can combine it with other vector operations like dot product and cross product, and that leads to the concepts of divergence and curl, respectively. De nition 1. We de ne the divergence of a ...
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