My Math Forum Oriented curve

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 September 19th, 2017, 08:48 AM #1 Member   Joined: May 2017 From: Slovenia Posts: 89 Thanks: 0 Oriented curve When is a curve oriented? And why do we orient curves?
 September 19th, 2017, 09:33 AM #2 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 The graph of the function y= x is the straight line through (0, 0) and (1, 1). That line can be "oriented" so that it goes from (0, 0) to (1, 1) or it can be oriented so that t goes from (1, 1) to (0, 0). Similarly, the unit circle, $\displaystyle x^2+ y^2= 1$ can be oriented "clockwise" or "counter-clockwise". There are many reasons why one would want to work with "oriented" curves. One is integration. In Calculus I we learn to integrate from a to b. In Calculus II or III, where we are dealing with functions of more than one variable, we might want to integrate f(x,y) along the line y= x from (0, 0) to (1, 1) or vice-versa. We might want to integrate f(x,y) around the unit circle "clockwise" or "counter-clockwise". Last edited by Country Boy; September 19th, 2017 at 10:11 AM.
September 19th, 2017, 09:44 AM   #3
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Quote:
 Originally Posted by Country Boy The graph of the function y= x is the straight line through (0, 0) and (1, 1). That line can be "oriented" so that it goes from (0, 0) to (1, 1) or it can be oriented so that t goes from (1, 1) to (0, 0). Similarly, the unit circle, $x^2+ y^2= 1$ can be oriented "clockwise" or "counter-clockwise". There are many reasons why one would want to work with "oriented" curves. One is integration. In Calculus I we learn to integrate from a to b. In Calculus II or III, where we are dealing with functions of more than one variable, we might want to integrate f(x,y) along the line y= x from (0, 0) to (1, 1) or vice-versa. We might want to integrate f(x,y) around the unit circle "clockwise" or "counter-clockwise".
And if we have scalar field and vector field.. in the vector field we need to orient the curve because the quantities that we measure in vector field have direction? So the curve has to be also oriented?
In my notebook it says:

Orientation of the curve is continuous vector function r: K-R^3 for which r(A) is unit tangent vector on K for every A E K.

what that means?

September 19th, 2017, 10:14 AM   #4
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 Originally Posted by sarajoveska And if we have scalar field and vector field.. in the vector field we need to orient the curve because the quantities that we measure in vector field have direction? So the curve has to be also oriented?
It is not a matter of the curve "having" to be oriented. The vector field gives it a "natural" orientation.

Quote:
 In my notebook it says: Orientation of the curve is continuous vector function r: K-R^3 for which r(A) is unit tangent vector on K for every A E K. what that means?
Does your notebook say what "K" is?

September 19th, 2017, 10:45 AM   #5
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Quote:
 Originally Posted by Country Boy It is not a matter of the curve "having" to be oriented. The vector field gives it a "natural" orientation. Does your notebook say what "K" is?
K is the curve.

 September 19th, 2017, 12:57 PM #6 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 So your function assigns, to every point on curve, K, a vector, a unit tangent vector to K at that point. At every point on a smooth curve, there are two unit tangent vectors pointing in opposite directions. The two different directions assign the two orientations.

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