5 Magnitude: slope along this maximal direction For a given |dr|, the change in scalar function f(x,y,z) is maximum when:=> f is a vector along the direction of maximum rate of change of the functionMagnitude: slope along this maximal direction
6 => df = 0 for small displacements about the point (x0,y0,z0) If f = 0 at some point (x0,y0,z0)=> df = 0 for small displacements about the point (x0,y0,z0)(x0,y0,z0) is a stationary point of f(x,y,z)
7 The Operator is NOT a vector, but a VECTOR OPERATOR Satisfies: Vector rulesPartial differentiation rules
8 can act: On a scalar function f : f GRADIENT On a vector function F as: . FDIVERGENCEOn a vector function F as: × FCURL
9 Divergence of a vector is a scalar. .F is a measure of how much the vector F spreads out (diverges) from the point in question.
10 Physical interpretation of Divergence Flow of a compressible fluid:(x,y,z) -> density of the fluid at a point (x,y,z)v(x,y,z) -> velocity of the fluid at (x,y,z)
11 (rate of flow in)EFGH(rate of flow out)ABCDZXYdydxdzADCBEFHG
12 Net rate of flow out (along- x) Net rate of flow out through all pairs of surfaces (per unit time):
13 Net rate of flow of the fluid per unit volume per unit time: DIVERGENCE
14 Curl of a vector is a vector ×F is a measure of how much the vector F “curls around” the point in question.
15 Physical significance of Curl Circulation of a fluid around a loop:Y3241XCirculation (1234)
16 Circulation per unit area = ( × V )|z z-component of CURL
17 Curvilinear coordinates: used to describe systems with symmetry. Spherical coordinates (r, , Ø)
26 Fundamental theorem for gradient We know df = (f ).dlThe total change in f in going from a(x1,y1,z1)to b(x2,y2,z2) along any path:Line integral of gradient of a function is given by the value of the function at the boundaries of the line.
32 Fundamental theorem for Divergence Gauss’ theorem, Green’s theoremThe integral of divergence of a vector over a volume is equal to the value of the function over the closed surface that bounds the volume.
33 Fundamental theorem for Curl Stokes’ theoremIntegral of a curl of a vector over a surface is equal to the value of the function over the closed boundary that encloses the surface.