1 / 19

Vector Calculus

Vector Calculus. Line Integral of Vector Functions. d l = dx i + dy j + dz k. For a closed loop, i.e. ABCA ,. = circulation of P around L. Line given by L( x ( s ), y ( s ), z ( s )), s = parametric variable.

sawyer-simon
Download Presentation

Vector Calculus

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. Vector Calculus

  2. Line Integral of Vector Functions dl = dxi +dyj +dzk For a closed loop, i.e. ABCA, = circulation ofP aroundL Line given byL(x(s), y(s), z(s)),s = parametric variable Always take the differential elementdlas positive and insert the integral limits according to the paths!!!

  3. Example ForF = yi –xj, calculate the circulation ofFalong the two paths as shown below. Solution: dl = dxi +dyj +dzk Along pathC2

  4. Example - Continue Along pathC1 Using xas the parametric variable, the path equations are given as Therefore, and

  5. Example - Continue The vector field defined by F in a given domain is non-conservative.The line integral is dependent on the integration path! is work done on an object along path C ifF= force !! Is the static Electrical field conservative? Yes, because the work done when we move a charge from one point to another is independent of the path but determined by the potential difference between these two points.

  6. Surface Integral Surface integral or the flux ofP across the surfaceSis is the outward unit vector normal to the surface. For closed surface, = net outward flux ofP.

  7. Example If F = xi + yj + (z2 – 1)k, calculate the flux ofFacross the surface shown in the figure. Solution:

  8. Volume Integral Evaluation : choose a suitable integration order and then find out the suitable lower and upper limits forx, y and zrespectively. Example: LetF = 2xzi –xj + y2k.Evaluate where Vis the region bounded by the surfacex = 0, x = 2, y = 0, y = 6, z = 0, z = 4.

  9. Volume Integral Solution: In electromagnetic, =Total charge within the volume where v = volume charge density (C/m3)

  10. Scalar Field Every point in a region of space is assigned a scalar value obtained from a scalar functionf(x, y, z), then a scalar fieldf(x, y, z) is defined in the region, such as the pressure in atmosphere and mass density within the earth, etc. Partial Derivatives Mixed second partials

  11. Gradient Del operator Gradient Gradient characterizes maximum increase. If at a pointPthe gradient offis not the zero vector, it represents the direction of maximum space rate of increase infat P.

  12. Example Given potential functionV = x2y + xy2 + xz2, (a) find thegradient ofV, and (b) evaluate it at (1, -1, 3). Solution: (a) (b) Direction of maximum increase

  13. Vector Field Electric field:E = E(x, y, z), Magnetic field :H = H(x, y, z) Every point in a region of space is assigned a vector value obtained from a vector functionA(x, y, z), then a vector field A(x, y, z) is defined in the region. R(t1, t2) = acos t1i + asin t1j + t2k

  14. Divergence of a Vector Field Representing field variations graphically by directed field lines - flux lines

  15. Divergence of a Vector Field The divergence of a vector field A at a point is defined as the net outward flux of A per unit volume as the volume about the point tends to zero: It indicates the presence of a source (or sink)!  term the source as flow source. Anddiv A is a measure of the strength of the flow source.

  16. Divergence of a Vector Field In rectangular coordinate, the divergence ofA can be calculated as For instance, ifA = 3xzi + 2xyj –yz2k, then div A = 3z + 2x– 2yz At (1, 2, 2), div A = 0; at(1, 1, 2),div A = 4, there is a source; at (1, 3, 1), div A = -1, there is a sink.

  17. Curl of a Vector Field The curl of a vector fieldAis a vector whose magnitude is the maximum net circulation ofAper unit area as the area tends to zero and whose direction is the normal direction of the area. It is an indication of a vortex source, which causes a circulation of a vector field around it. Water whirling down a sink drain is an example of a vortex sink causing a circulation of fluid velocity. If Ais electric field intensity, then the circulation will be an electromotive force around the closed path.

  18. Curl of a Vector Field In rectangular coordinate, curl A can be calculated as

  19. Curl of a Vector Field Example: If A = yzi + 3zxj + zk, then

More Related