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Summary of Multivariable Calculus MVC Lecture 9 for MATH2310 Engineering Calculus $7.99   Add to cart

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Summary of Multivariable Calculus MVC Lecture 9 for MATH2310 Engineering Calculus

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Summary of Lecture 9 for the multivariable calculus, MVC, component of MATH2310, calculus for engineering and science, at the University of Newcastle.

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  • July 1, 2023
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MATH2310 – MVC Lecture 9


Curl:
There are two operations on vector fields that are fundamental to the study of fluid
flow, and electricity and magnetism. Each operation resembles a differentiation, but
one produces a vector field, and the other produces a scalar field.
If we write:

∇= ⟨ ∂ ∂ ∂
, ,
∂x ∂ y ∂z ⟩
so that

∇f= ⟨ ∂ ∂ ∂
, ,
∂x ∂ y ∂z
f ⟩
then we can view ∇ as a vector operation that resembles differentiation.

If F is a vector field on R3 , the curl of F is given by:

curl ( F )=∇ × F

If F=⟨ P , Q , R ⟩ ,


( ∂∂ Ry − ∂∂Qz ) i,( ∂∂ Pz − ∂∂ Rx ) j ,( ∂Q
curl ( F )=
∂x ∂y)

∂P
k

If f is a function of three variables, with continuous second-order partial derivatives,
then curl ( ∇ f ) =0

If a vector field, F , is conservative, then:
curl ( F )=0

Conversely, if a vector field, F , is not conservative, then:
curl ( F ) ≠ 0

Similarly, if F is a vector field whose domain is R3, whose component functions have
continuous partial derivatives, and if curl ( F )=0 , then F is a conservative vector field.

This operation is the curl because it is associated with rotations. For instance,
suppose F represents velocity in a fluid flow, then particles near (x , y , z) tend to
rotate or curl around vector curl (F ( x , y , z ) ). The magnitude of the vector indicates
how quickly the particles do so.

If curl ( F )=0 at a point P , then P is free from rotation at P , and F is called irrotational
at P . In other words, there is no ‘whirlpool’ or ‘eddy’ at P .

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