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Gradient of a scalar field

This has the form of a scalar product of the differential displacement dr = idx+jdy + lidz with the gradient vector [Pg.217]

When d r happens to lie within one of the surfaces of constant 0, then d(p = 0, which implies that the gradient V0 at every point is normal to the surface 0 = const containing that point. This is shown in Fig. 11.10, with [Pg.217]

FIGURE 11.10 Gradients V0 of a scalar field shown by arrows. The gradient at every point x, y, and z is normal to the surface of constant 0 in the direction of maximum increase of 0(x, y, z). [Pg.217]

The change in a scalar field 0(r) along a unit vector u is called the [Pg.218]

A finite change in the scalar field (r) as r moves from ri to rz over a path C is given by the line integral [Pg.218]


If V(r, x) were a known function, this linear expansion could be used to determine how the velocity varies for short intervals of time and in any arbitrary short spatial direction dx. In a Taylor-series expansion of a scalar field, it is often conventional to post-multiply by the dx. Since the gradient of a scalar field is a vector and because the inner product of two vectors is commutative, the order of the product is unimportant. However, because of the tensor structure of the gradient of a vector field, the pre-multiply is essential. [Pg.26]

In this case the pressure is eliminated altogether, since by vector identity, the curl of the gradient of a scalar field vanishes. From the definition of vorticity, Eq. 2.103, a simple diffusion equation emerges for the vorticity... [Pg.135]

So Hs is a irrotational field and can therefore be expressed as gradient of a scalar field % called magnetic potential. [Pg.106]

The gradient of a scalar field A(x, y,z) is a vector field, each point of which gives the magnitude and direction of the largest change in the scalar field ... [Pg.379]

The divergence of the gradient of a scalar field occurs in several fundamental equations of electromagnetism, wave theory, and quantum mechanics. In Cartesian coordinates. [Pg.221]

Equation (14) is known as the curl-condition and derives from the elementary fact that the vector field Fi2(Q) should be curl-free if expressible as the gradient of a scalar field according to Eq. (13), since one has ... [Pg.181]

Gradient of a scalar The gradient of a scalar field p is a vector defined by Vp, or Qp/dxi. [Pg.151]

The differential operators encountered often in the description of the physical properties of solids are the gradient of a scalar field V,4)(r), the divergence of a vector field Vr F(r), the curl of a vector field x F(r), and the laplacian of a scalar field V vector field is simply the vector addition of the laplacian of its components, V F = + V F z). These operators in three dimensions are... [Pg.644]

A number of the force fields occurring in physics can be described as gradients of a scalar field x, y, z). With negative sign, the scalar field denotes the so-called potential field belonging to the actual force field. Prom the above, it is seen that the work with a given displacement in this kind of force field is independent of the path chosen in the field these kinds of force fields, therefore, will always be conservative fields. [Pg.265]

Scalar, vector, and tensor fields. The gradient of a scalar field 0(x) is denoted by V0 and is the veetor defined by ... [Pg.7]


See other pages where Gradient of a scalar field is mentioned: [Pg.440]    [Pg.154]    [Pg.85]    [Pg.154]    [Pg.687]    [Pg.1160]    [Pg.217]    [Pg.217]    [Pg.347]    [Pg.716]    [Pg.181]    [Pg.687]    [Pg.523]    [Pg.1443]    [Pg.11]    [Pg.267]   
See also in sourсe #XX -- [ Pg.11 ]




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