1 / 10

Some Theorems

Learn about Divergence Theorem for solid regions with vector fields and Stokes' Theorem for oriented surfaces with piecewise-smooth curves in 3D space. Explore examples with detailed evaluations.

gmixson
Download Presentation

Some Theorems

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Some Theorems Thm. Divergence Theorem Let E be a solid region and let S be the boundary surface of E. Let F be a vector field whose component functions have continuous partial derivatives on an open region containing E. Then

  2. Ex. Evaluate where F = xi + 2yj + zk and S is the surface of the solid bounded by 2x + 2y + z = 6 and the coordinate planes.

  3. Ex. Find the flux of the vector field F = zi + yj + xk over the unit sphere.

  4. Ex. Evaluate where F = xyi + (y2 + exz)j + ln(xy)k and S is the surface of the region bounded by z = 1 – x2, z = 0, y = 0, and y + z = 2

  5. Thm. Stokes’ Theorem Let S be an oriented piecewise-smooth surface that is bounded by a simple, closed, piecewise-smooth curve C. Let F be a vector field whose components have continuous partial derivatives on an open region in 3 that contains S. Then

  6. Ex. Evaluate where F = -y2i + xj + z2k and C is the curve of intersection of the plane y + z = 2 and the cylinder x2 + y2 = 1.

  7. Ex. Evaluate where F = -y2i + zj + xk and C is the boundary of the portion of 2x + 2y + z = 6 that lies in the first octant.

  8. Ex. Evaluate where F = xzi + yzj + xyk and S is the part of the sphere x2 + y2 + z2 = 4 that lies inside the cylinder x2 + y2 = 1 and above the xy-plane.

  9. Line Integrals • If F(x,y) is conservative: • We can change the path • if the path is closed • If C is closed: • (Green’s Theorem)

  10. Surface Integrals • If S is closed: • (Divergence Theorem) • If C is the edge of S: • (Stokes’ Theorem)

More Related