## Derivative WRT to a Scalar

4 The derivative of a vector p(u) with respect to a scalar u, Fig. 3.2 is defined by dp = lim p(u + Au) p(u) du Au^o Au

5 If p(u) is a position vector p(u) = x(u)i + y(u)j + z(u)k, then dp dx. ^ dy . ^ dz ^ ^ ^

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■ Scalar and Vector Fields

ContourPlot [Exp [- ¡x'2 + y^2)], {x, -2, 2}, {y, -2, 2}, ContourShading -> False]

- ContouiGraphics -

Plot3D [Exp [-¡xA2 + y^2 )] , {x, -2, 2}, {y, -2, 2}, FaceGrids -> All]

- SurfaceGraphics -  Figure 3.1: Examples of a Scalar and Vector Fields Figure 3.2: Differentiation of position vector p is a vector along the tangent to the curve.

6 If u is the time t, then dp is the velocity

7 In differential geometry, if we consider a curve c defined by the function p(u) then ^p is a vector tangent ot C, and if u is the curvilinear coordinate s measured from any point along the curve, then pp is a unit tangent vector to c T, Fig. 3.3. and we have the following relations Figure 3.3: Curvature of a Curve

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