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Gradient, Divergence and C i, , gradient, divergence and curl are three fundam fy, sations. The gradient transforms a scalar function into a, ferent ion. The divergence and curl operations transform a vector |, gto a scalar and a vector function, respectively,, 41 Scalar and Vector Fields, , Definition 1.A real-valued function 9 (x, y, 2) of three salar variables fa, , y,zis known as a scalar field. ai y /, , ¥, i, , For example, © (x, y, Z) = r+ y +z is a scalar field., Definition 2. A vector function A (x,y,z) of three scalar variables —, x,y,z given by, A (x, y, 2) =Aj (x, y, Zi + Az (x, y, Z) J + Ag (%, y, z) kK, iscalled a vector field., , For example, A (x, y, 2) =x” y~ i+y? 2 j+z xk is a vector field,, , Given a vector field A (x, y,z), we can draw from each point (x, y, z) 4, ofits domain the vector arrow representing the vector A (x, y, z)., , Definition 3.A vector field A = Ay i+ Az j +A, kis called coutinmare, feach of the components Aj, Az, Ay is continuous., , Now we define some vector differential Operstigns: thst ia, 52. Vector Differential Operations Se, , Definition 4. (Del Operator), , The vector diff rential operator del, written V, is de, , , , , Vasiea j+2 zk or Vaiss

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50, , , , , , , , , , , , , , , , , Definition 6. (Divergence), The divergence of a vector function, A (x,y,z) =A) y, 2) i+ Ar (XY, Zz) J+A, where A,, Az and A; have first order partial derivatives in some, V. A ordiv A, is defined as, divA=V A= St a Be :, Tt may be noted that div A is a scalar function and V. A#A, i) Definition 7. (Curl), ; The curl of a vector function ., A (x,y. 2) =A) (xy, i+ Ad (2) 5 +43 OY, 2)I, , where Ay, Az and A, have first order partial derivatives in some regi, 1 Vx Aor curl A, is defined by, , +, , , , Lojane, , curlA=VxA= aoe, , ox dy az, Pa Ay Ap Ay, L Equivalently,, cuan( SS l(t +(e, It may be noted that curl A is a vector., Remarks, , 1.V.A=divA andVxA=curlA., In other words, del det = div and del cross = curl., 2. Ifr=xi+yj+zk, then divr=3 and curl r=0., , , , , A a 0 a, We see that ara O)+57@)= I+.