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# Stokes example part 2

Parameterizing the surface. Created by Sal Khan.

## Want to join the conversation?

• i dont understand why is sal using the unit circle, isnt the intersection of the shape of an ellipse? • Because all the X and Y points will lie within the Unit Circle. He then completed the vector expressing the Z points in terms of the X and Y points. No Z he was concerned with "lived" outside the projection of the Ellipse on the X,Y planes: the shadow of the ellipse on the XY planes is a unit circle
• So when paramterizing for stokes theorem, you have to considered every point inside the region, but for just the line integral, its just the edges and not inside the region for example? like its a disk, not a circle, but for fevaluating line integral directly, its a cirlce, not a disk. Can someone please helpme out on this i have a final in 6 days on this. • Before evaluating any surface integral, one needs to take into account every point at which you are computing the integral by parameterizing the surface using two independent variables. Before evaluating a line integral, one must take every point covered along the path using a one variable parameterization. Creating a vector representation of either a surface or closed path is often the most difficult part of applying Stokes Theorem. From my experience, it takes practice to develop that skill to a second nature degree. Usually, once you have conceived of a suitable parameterization, the problem involves calculating a single, double, or triple integral. I hope this helps.
• Why should we split the surface into three surfaces and separately apply stokes law like in the previous video where he calculated the surface integral... how can we parametrize taking it as one surface , wont taking a variable radius r and a variable angle trace out a solid and not a surface.? And is this surface open at the top or the base? • The vector-valued function that is created in this video does not define the surface S but rather the region bounded by the curve c. This occurs because z is defined explicitly as a function of y and therefore can only take on values sitting on the plane y+z=2. This is also the reason that the function does not define a solid, i.e. z cannot take on values in the range: [0,y+z=2).
(1 vote)
• Why is n = S_r x S_theta dr d(theta)? • I am confused how at the end you came up with "little s". Not the variable but the equation. • I'm still struggling to understand when to use when to use i-j+k vs. i+j+k. Are there any videos that explain when it's appropriate to use one verses the other? Thanks!
(1 vote) • this may be a trivial question, but do I , j and k represent the unit vectors indirection of x, y, z?
(1 vote) • If we were to use this parameterization, could we build a 3-d Knot? This seems like a really good way to describe a basic set of them. like a clove hitch, which I feel can be done like a torrus. 