1

Vector Calculus

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Figure 3.1 Differential elements in the right-handed Cartesian coordinate system.

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Figure 3.2 Differential normal surface areas in Cartesian coordinates: (a) dS dy dz ax, (b) dS dx dz ay, (c) dS dx dy az.

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Figure 3.3 Differential elements in cylindrical coordinates.

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Figure 3.4 Differential normal surface areas in cylindrical coordinates: (a) dS d dz a, (b) dS d dz a, (c) dS d d az.

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Figure 3.5 Differential elements in the spherical coordinate system.

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Figure 3.6 Differential normal surface areas in spherical coordinates: (a) dS r2 sin d d ar,(b) dS r sin dr d a, (c) dS r dr d a.

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Figure 3.7 For Example 3.1.

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Figure 3.8 For Practice Exercise 3.1 (and also Review Question 3.3).

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Figure 3.9 Path of integration of vector field A.

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Figure 3.10 The flux of a vector field A through surface S.

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Figure 3.11 For Example 3.2.

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Figure 3.12 For Practice Exercise 3.2, L is a closed path.

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Figure 3.13 Gradient of a scalar.

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Figure 3.14 For Example 3.5; plane of intersection of a line with an ellipsoid.

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Figure 3.15 Illustration of the divergence of a vector field at P: (a) positive divergence, (b) negative divergence, (c) zero divergence.

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Figure 3.16 Evaluation of A at point P(x0, y0, z0).

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Figure 3.17 Volume v enclosed by surface S.

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Figure 3.18 For Example 3.7.

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Figure 3.19 Contour used in evaluating the x-component of A at point P(xo, yo, zo).

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Figure 3.20 Illustration of a curl: (a) curl at P points out of the page, (b) curl at P is zero.

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Figure 3.21 Determining the sense of dl and dS involved in Stokes’s theorem.

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Figure 3.22 Illustration of Stokes’s theorem.

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Figure 3.23 For Example 3.9.

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Figure 3.24 Typical fields with vanishing and nonvanishing divergence or curl.

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Figure 3.25 For Review Question 3.1.

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Figure 3.26 For Review Question 3.2.

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Figure 3.27 For Review Question 3.8.

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Figure 3.28 For Problem 3.7.

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Figure 3.29 For Problem 3.31.

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Figure 3.30 For Problem 3.32.

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Figure 3.31 For Problem 3.33.

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Figure 3.32 Volume in form of ice cream cone for Problem 3.38.