Understanding Curl: How to Evaluate Line Integrals and Taylor Expansions

[A New Learner]

Hello All ..
How are you ? I hope you fine

Our professor taught as about the meaning of curl , but I was totally confused about it , especially when he used Taylor expansion of two variables and line integrals

It’s like this

Sorry for the very bad diagram in attachments , where delta means ∆
∮▒(V.) ⃗ dλ ⃗
= ∮▒〖(Vxdx+Vydy)〗
Where Vy and Vx are velocity vector component
Then he expanded Vx and Vy about p
Vx = Vx(p) + ∂Vx/∂x(p) (x-xo)+ ∂Vy/∂y(p) (y-yo)
Vx = Vx(p) + ∂Vx/∂x(p) (x-xo)+ ∂Vy/∂y(p) (y-yo)

I want hinge explanation please about the above point ..
Then he asked us to evaluate the line integral for
∮▒(Vxdx+Vydy)
( By the expanded bionomials ) , the linear integral is from A TO B then form B TO C the from C TO D
He said the result should be
(∂Vy/∂y- ∂Vx/∂x)∆x∆y= Curl (V ⃗ )z
But HOW
Please .. I want aalso a hinge explanation for this .. and thanks

 

[tiny-tim]

Welcome to PF!

Hello A New Learner! Welcome to PF!

Hello All ..
How are you ? I hope you fine

I'm very well, thankyou.

…Then he expanded Vx and Vy about p
Vx = Vx(p) + ∂Vx/∂x(p) (x-xo)+ ∂Vy/∂y(p) (y-yo)
Vx = Vx(p) + ∂Vx/∂x(p) (x-xo)+ ∂Vy/∂y(p) (y-yo)
…
(∂Vy/∂y- ∂Vx/∂x)∆x∆y= Curl (V ⃗ )z

No, they should be …

Vx = Vx(p) + ∂Vx/∂x(p) (x-xo)+ ∂Vx/∂y(p) (y-yo)
Vy = Vy(p) + ∂Vy/∂x(p) (x-xo)+ ∂Vy/∂y(p) (y-yo)

(and although you can use Taylor's theorem to get these, you don't need to … they follow directly from the definition of the derivative of a function)

(∂Vy/∂x- ∂Vx/∂y)∆x∆y= Curl (V ⃗ )z

 

[A New Learner]

tiny-tim

thank you so much for your help .. and I'm sorry for the wrong writing of the equation ..
but I'm still wondering how to evaluate these linear integerals

Best Wishes ..

 

[tiny-tim]

tiny-tim

thank you so much for your help .. and I'm sorry for the wrong writing of the equation ..
but I'm still wondering how to evaluate these linear integerals

Best Wishes ..

Hi A New Learner!

To start you off:

you're going round all four sides of that rectangle …

so the first side will be from xo, yo to xo + ∆x, yo …

so integrate Vxdx + Yydy along that, ignoring any terms with more than two derivatives, and then the same along the other three sides.