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Sec 17.5 Curl and Divergence. We define the vector differential operator (“del”) as Del has meaning when it operates on a scalar function f to produce the gradient of f :. Definition : The Curl of a vector field F.
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Sec 17.5 Curl and Divergence We define the vector differential operator(“del”) as Del has meaning when it operates on a scalar function f to produce the gradient of f :
Definition: The Curl of a vector field F If F = P(x,y,z)i + Q(x,y,z)j + R(x,y,z)k isa vector field in 3-space, and the partial derivatives of P, Q, and R all exist. The curl of Fis the vector field defined by
Theorem: If f is a function of three variables that has continuous second-order partial derivatives, then Theorem: Let F = P(x,y,z)i + Q(x,y,z)j + R(x,y,z)k be a vector field in 3-space, and P, Q, and Rhave continuous partial derivatives. If curl F = 0, then F is a conservative vector field.
Definition: The Divergence of a vector field F If F = P(x,y,z)i + Q(x,y,z)j + R(x,y,z)k is a vector field in 3-space, and the partial derivatives exist, then the divergence of F is the function of three variables defined by Theorem: If F = P i+ Q j+ R k , and P, Q,and R have continuous second-order partial derivatives, then div curl F = 0. [ Note: curl F is a vector field but div F is a scalar field.]
