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Vector Calculus sector of 6.2 17. D is always on the left as we travel along C (C is the path of D)

What is the meaning of the above statement? I can't understand ;(

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I think it would be best to have a lot more context, and especially a scanned image of the text you're using. I'll make some assumptions:

  1. D refers to the domain of the function on which the original integral was supposed to be done
  2. C is a curve that encompasses D

If these two statements are correct, then it means the following:

Let us draw a curve, C, around the domain, D. Imagine that the domain is like a lake, and the curve is a road that goes around the perimeter of that lake. Imagine yourself driving a car along that road. You can choose to go in one of two directions - either clockwise or counter-clockwise. If you travel counter-clockwise along the road, and you look out of a window on the left hand side of the car, you will see the lake. If you look out of the right hand side, you will not see the lake.

The textbook is saying that you should move along your curve in a counter-clockwise direction. Mathematically, if the curve is parameterized, then you need to choose your parameterization so that the position vector along the curve makes a rotation counter-clockwise.

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