Gradient Scalar

16 The gradient of a scalar field g(x) is a vector field Vg(x) such that for any unit vector v, the directional derivative dg/ds in the direction of v is given by d9 — — = Vg^v ds

3.4 Gradi

■ Divergence of a Vector

<< Calculus'VectorAnalysis1 V = (xA2z, -2yA3zA2, xy^2z}; Div[V, Cartesian[x, y, z]] —6 z2 y2 + xy2 +2 xz << Graphics,PlotField3D1

PlotVectorField3D[(x*2 z, -2yA3zA2, xy^2 z} , (x, -10, 10}, (y, -10, 10}, (z, -10, 10}, Axes-> Automatic, AxesLabel -> ("X", "Y", "Z"}]

- Graphics3D -

Div[Curl[V, Cartesian[x, y, z]], Cartesian[x, y, z 0

Figure 3.8: Mathematica Solution for the Divergence of a Vector

- Graphics3D -

Div[Curl[V, Cartesian[x, y, z]], Cartesian[x, y, z 0

Figure 3.8: Mathematica Solution for the Divergence of a Vector where v = dp We note that the definition made no reference to any coordinate system. The gradient is thus a vector invariant.

17 To find the components in any rectangular Cartesian coordinate system we use dp dxi v dg ds ds ds dg dxi dxi ds which can be substituted and will yield or

and note that it defines a vector field.

is The physical significance of the gradient of a scalar field is that it points in the direction in which the field is changing most rapidly (for a three dimensional surface, the gradient is pointing along the normal to the plane tangent to the surface). The length of the vector ||V#(x)|| is perpendicular to the contour lines.

19 Vg(x)n gives the rate of change of the scalar field in the direction of n.

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