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He keeps talking about a line integral around a closed-loop, ANDthe integral over an open surface.What is the definition of a line integral around aclosed-loop?What...

Question

He keeps talking about a line integral around a closed-loop, ANDthe integral over an open surface.What is the definition of a line integral around aclosed-loop?What is the definition of an open surface (compared to a closedsurface)?

He keeps talking about a line integral around a closed-loop, AND the integral over an open surface. What is the definition of a line integral around a closed-loop? What is the definition of an open surface (compared to a closed surface)?



Answers

Discuss the similarities and differences between the definition of a surface integral and the definition of a double integral.

The Divergence Theorem says that this surface, integral on the left here, is equal to this volume into role on the right here. Let's take a closer look at the surface integral on the left. What does it mean? Well, f dot n picks out the component of our vector field going straight in or out across the surface at a point. There is a typical blob of a surface, and here are some lines of our vector field. You might have lines coming out another place. You might have lines coming in an is three unit normal vector perpendicular to the surface. Right at that point, and taking the DOT product with our field picks out Ghias, the component of the field, which is straight, perpendicular to the surface and the integral adds up all of those little ends and outs all the way across the surface around got. Imagine. That's three dimensional, even though I only drew two dimensional, and when you add them all up, you get what we call the flux. The flux is the net flow of our vector field in or out across the surface, so that's the left side of the divergence theorem. This surface, integral, is the flux of the vector field. Across the surface

Problem number one, the Integral After the X is aligned integral. It represents the net circulation on the closed simple piece wise, smooth oriented curve C that forms the boundary off the oriented surface s.

So in Stokes is through him Meaning of this Andrew All Hugh simply describes the line integral over the contra. See of us dot Thea dio goes along the contract. So, for example, if we look at this graph and say this red outline, um is ah, control this integral hell would describe the line integral if we're true booked if we are to traverse this shape counterclockwise.

In Stokes is through the line, integral over the contour. See, of s dot de, uh, d r Goes along The contour describes the counterclockwise circulation of a vector field along a closed cove sea and all the wounds. It describes the tendency of a fluid to circulate around a boundary coat. So, for example, if we look at this surface, you the red boundary. Cool. If we look at ah first interval from Stokes, assume it simply describes the tendency of the fluid to circulate around this CUF.


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