Scalar and vector fields

A scalar field is a function defined on the three-dimensional space coordinates and possibly along the time axis. An example is the temperature field T(x, y, z; t) = T (r, t), where the position vector r is defined by r = x i + y j + z k (9.17)

and i, j, k are unit vectors pointing along the x,y and z axes (see Figure 9.5). A small increment in r is denoted as 1

Figure 9.5 Schematic diagram of a position vector r whose components are x, y and z.

1 Here we replace the small values Ax, Sx, etc., with infinitesimals dx, etc., with the approximate sign ^ replaced by the equality sign =. This means that in this notation second-order quantities such as (dx)2 are neglected (set to zero) when additive to first-order terms. While this operational shortcut might cause some to cringe, it should not disturb the flow of our story.

Figure 9.6 Schematic diagram of a position vector r and an increment of it dr.

Consider the estimation of the temperature field at the point r + dr (Figure 9.6), given that we know its value at the point r, namely T (r):

We may use the first two terms of the Taylor expansion:

d T d T dT T (x + dx, y + dy, z + dz) = T (x, y, z) + — dx + — dy + — dz. (9.19)

dx dy dz

We can also write this as a dot product:

T(r + dr) = T(r) + dr ■ VT(r). After substituting dT = T(r + dr) — T(r), we obtain dT = dr ■ VT [differential of a scalar field].

The vector VT (r) is called the gradient ofT. We will use the modern notation VT to denote the gradient (in some older texts it is denoted grad T).

[gradient of a scalar field].

The gradient is a vector field. At each point in space r it has an associated length and direction.

If you want to know the rate of change of the field in a particular direction, say along the direction defined by the unit vector, n, it can be found by defining2 the vector increment dr to be n ds where ds is an infinitesimal distance and n defines the direction along which the increment is to be taken. Using (9.21) we can write dr

[derivative in the direction n]. (9.23)

This derivative taken along the direction of the specified unit vector n is called the directional derivative, and is often given the notation d T/dn as the rate of change of T along a certain direction, defined by the unit vector n. The conventional notation for the directional derivative is:

[directional derivative]. (9.24)

If n lies in the tangent plane to an isothermal (still thinking of the scalar field as temperature) surface, the directional derivative vanishes since there is no change in any direction lying in this plane. This means that the component (projection) of the gradient vector tangent to the isothermal surface vanishes. The gradient vector is perpendicular to isothermal surfaces (in general so-called level surfaces). This can be seen for a fixed gradient vector VT. Just vary the unit vector in all directions. The lengths of n and VT are fixed, so the maximum occurs when the angle between n and VT is zero (cos 0n,VT = 1), in other words when n is parallel to VT.

Example 9.7 Consider the field

Find the gradient as a function of x and y.

Answer:

VT(x, y) = — nT0 (2sin2nx cos nyi + cos2nx sin nyj). (9.26) See the contour map in Figure 9.7. □

Example 9.8 Find the directional derivative of the field in the last example in the direction n = (1/V2)(i + j) (this is a unit vector in the x-y plane directed 45° above the x-axis).

Answer: Take the dot product of n with the gradient:

—n To n • VT = —— (2 sin 2nx cos ny + cos 2nx sin ny). □

Remember that the reader has the power to choose dr, its tiny length and direction.

Figure 9.7 Contour map of a field T(r) = cos2nx cos ny showing constant T lines. Arrows indicate direction of the gradient vector evaluated at the points where the arrows originate.

Figure 9.7 Contour map of a field T(r) = cos2nx cos ny showing constant T lines. Arrows indicate direction of the gradient vector evaluated at the points where the arrows originate.

Figure 9.8 A small rectangular parallelepiped of side widths dx, dy, dz, indicating the pressure forces on the sides perpendicular to the x-axis.

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