PDF-Gradient Divergence and Curl in curvilinear coordinates

inCurvilinearCoordinates AlthoughcartesianorthogonalcoordinatesareveryintuitiveandeasytouseitisoftenfoundmoreconvenienttoworkwithothercoordinatesystemsBeingabletochangeallvariablesandexpressioninvol. Curl of a gradient is 8711 x y z 8706x 8706y 8706z 8706t 8706x 8706t 8706y 8706t 8706z x 8706y8706z 8706z8706y y 8706z8706x 8706x8706z z 87 The integral is performed through the volume where the currents are but as usual we can extend th eintegraltoallspaceforfreesincewhere is zero the contribution to the integral vanishes anyway Before starting computing derivatives let s be totally ex The distribution of temperature may be represented by drawing isothermal surfaces or contours connecting points of identical temperatures One can draw such contours for different temperatures If we are located at a point on one of these contours and So if PQR then div 8711 8706x 8706y 8706z PQR 8706P 8706x 8706Q 8706y 8706R 8706z Notice that div is a scalar Find div for each of the following vector 64257elds i xyyzxz ii yzxzxy iii where iv grad where is a function with continuous second der By the way the gradient of isnt always denoted sometimes its denoted grad As you know the gradient of a scalar eld is f x f x f x We can abstract this by leaving out the to get an operator x x x which when applied to yields This is called t It is important to remember that expressions for the operations of vector analysis are different in di64256erent coordinates Here we give explicit formulae for cylindrical and spherical coordinates 1 Cylindrical Coordinates In cylindrical coordinate e is identi64257ed with the vector that is obtained by translating to the point Thus every vector 64257eld on is uniquely determined by a function from Ra64257kul Alam IITG MA102 2013 brPage 3br Vector Fields Curl and Divergence Examples of vector The divergence of a vector field. The curl of a vector field. Their physical significance. Divergence of a vector field. Curl of a vector field.44.1 Divergence of a v Figure4.2:Lineintegral.InthediagramF(r)isavectoreld,butitcouldbereplacewithscalareldU(r).4.1.1Physicalexamplesoflineintegralsi)ThetotalworkdonebyaforceFasitmovesapointfromAtoBalongagivenpathCisgiven x2+y2;q=arctan(y=x):TheChainRuleenablesustorelatepartialderivativeswithrespecttoxandytothosewithrespecttorandqandviceversa,e.g. COORDINATE SYSTEMS. . RECTANGULAR or Cartesian. . CYLINDRICAL. SPHERICAL. Choice is based on symmetry of problem. Examples:. Sheets - RECTANGULAR. Wires/Cables - CYLINDRICAL. Spheres - SPHERICAL. Preclass0. Yes, I do!. Do you have an Iclicker to use for this term?. Press & HOLD power (blue light . flashes). Key in . AB. (OUR code for this room). Brief green Status flash confirms! . (Blue light steady). Week 2. Vector Operators. Divergence and . Stoke’s. Theorems. Gradient Operator. The gradient is a vector operator denoted . . . and sometimes also called “del.” It is most often applied to a real function of three variables. . Stokes's. theorem: . The . curl . of . A . is the rotational vector whose magnitude is the maximum circulation of A per unit area as the area tends to zero and whose direction is the normal direction of the area when the area is oriented so as to make the circulation maximum..