The Gradient of a Scalar1 A scalar c varies with position according to: c = x^2 + 4y^2. (E5.1.1) That is, surfaces of constant values of c are ellipses, two of which (for c = 4 and c = 16) are shown in Fig. E5.1. Evaluate the gradient of c at four representative points, A, B, C, and D on the ellips

Question

The Gradient of a Scalar [latex]^{1}[/latex]

A scalar [latex]c[/latex] varies with position according to:

[latex]c=x^{2}+4 y^{2} .[/latex] (E5.1.1)

That is, surfaces of constant values of [latex]c[/latex] are ellipses, two of which (for [latex]c=4[/latex] and [latex]c=16)[/latex] are shown in Fig. E5.1. Evaluate the gradient of [latex]c[/latex] at four representative points, A, B, C, and D on the ellipse for which [latex]c=16[/latex]. The coordinates of these points are [latex](-4,0),(4,0),(2, \sqrt{3})[/latex], and [latex](0,2)[/latex].

[latex]^{1}[/latex] For Examples 5.1-5.4, two-dimensional situations have been chosen because: (a) they are easily visualized, and (b) this text is largely confined to no more than two-dimensional problems.

Accepted Answer

The derivatives of [latex]c[/latex] with respect to the two coordinate directions, and the corresponding gradient are:

[latex]\frac{\partial c}{\partial x}=2 x, \quad \frac{\partial c}{\partial y}=8 y, \quad abla c=2 x \mathbf{e}_{x}+8 y \mathbf{e}_{y} .[/latex] (E5.1.2)

Table E5.1 Gradients ∇c and Magnitudes \|∇c\| at Four Points
Point	x	y	∇c	\|∇c\|
A	-4	0	-8[latex]e_{x} [/latex]	8
B	4	0	8[latex]e_{x} [/latex]	8
C	2	[latex]\sqrt{3}[/latex]	4[latex]e_{x} [/latex]+8[latex]\sqrt{3} e_{y} [/latex]	[latex]4\sqrt{3} [/latex]
D	0	2	16[latex]e_{y} [/latex]	16

The individual gradients and their magnitudes at the four points are given in Table E5.1. Observe that each gradient is pointing normally outwards from the ellipse, in the direction of increasing [latex]c[/latex], and that the magnitude of [latex] abla c[/latex] is greatest at point D on the [latex]y[/latex]-axis, where the ellipses are spaced most closely.

Question 5.1: The Gradient of a Scalar1 A scalar c varies with position ac...

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