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 

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 

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). 

The change in a scalar field 0(r) along a unit vector u is called the 

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 

---

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. 

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... 

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

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 ... 

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

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 ... 

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

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... 

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. 

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

---