Divergence Vector Calculus
Divergence in vector calculus measures the rate at which a vector field's intensity spreads out from a given point. It is a scalar quantity that represents the volume density of the outward flux of a vector field per unit volume. In practical terms, divergence is used to analyze fluid flow, electric and magnetic fields, and other phenomena in engineering and technology.
3 Key excerpts on "Divergence Vector Calculus"
eBook - ePub- John Newman, Vincent Battaglia, John Newman, Vincent Battaglia(Authors)
- 2020(Publication Date)
- Jenny Stanford Publishing(Publisher)
...Appendix A Vectors and Tensors You have seen that some of our equations, such as the equation of motion, have become quite lengthy. This frequently leads one to introduce vector notation, which has several advantages: 1. The equations become considerably more compact when written in vector notation. 2. The equations have significance independent of any particular coordinate system. 3. It is easier to grasp the meaning of an equation (after the vector notation becomes familiar). For the present, you may regard vector notation as a form of shorthand writing. However, it would be a good idea for you to develop an intuitive feel for the significance of some of the more common vector operations. A vector has both magnitude and direction and can be decomposed into components in three rectangular directions: The divergence of a vector field is This quantity is a scalar whose physical significance can be seen most easily from the continuity equation The mass flux is ρ v, showing the direction and magnitude of mass transfer per unit area, and ∇·(ρ v) represents the “rate of mass flowing away from a point.” Hence the name “divergence.” We might call −∇·(ρ v) the “convergence” of the mass flux r v. Then the equation of continuity says Rate of accumulation = Rate of convergence or net input. The curl of a vector field yields another vector. When v is the fluid velocity, Ω is known as the “vorticity,” which may be regarded as the angular velocity (radians/second) of a fluid element. This vector operation will be used very seldom (if at all) in this course. The gradient of a scalar field is a vector. The gradient of ρ shows the change of density with position. The direction of the gradient shows the direction of the greatest change, and the magnitude is the rate of change in this direction. The gradient of a vector field, on the other hand, is a tensor...
eBook - ePubFundamental Principles of Optical Lithography
The Science of Microfabrication
- Chris Mack(Author)
- 2011(Publication Date)
- Wiley(Publisher)
...The dot product is sometimes called the scalar product or the inner product. It represents the product of the length of one vector with the portion of the length of the second vector that lies in the same direction as the first. If the dot product of two nonzero vectors is zero, then those vectors must be perpendicular to each other. B.3 Curl In Cartesian coordinates, the curl of a vector field F is defined as (B.6) (The designation of F as a field simply reflects the notion that the value of F varies as a function of position in space.) The curl of F is sometimes expressed as curl F. The curl is often thought of as showing the vector’s rate of rotation at the point of evaluation. The curl of a vector field always points in a direction perpendicular to the vector at the point of evaluation. B.4 Divergence The divergence is an operator that produces a scalar measure of a vector field’s tendency to originate from or converge upon a given point (the point at which the divergence is evaluated). In Cartesian coordinates, the divergence of a vector field F is defined as (B.7) The divergence of F is sometimes expressed as div F. Physically, the divergence is often thought of as the derivative of the net flow of the vector field out of the point at which the divergence is evaluated. Divergence is a linear operator so that for two scalar constants a and b and two vectors U and V, (B.8) B.5 Gradient In Cartesian coordinates, the gradient of a scalar field g is defined as (B.9) The gradient of g is sometimes expressed as gradg. It is interesting to note that the dot product of the gradient of a function with a unit vector gives the slope of the function in the direction of the unit vector. For example, (B.10) giving what is sometimes called the directional derivative. B.6 Laplacian The Laplacian operator, equal to the divergence of the gradient, operating on some scalar field g, is given in Cartesian coordinates as (B.11) The Laplacian is a second-order differential operator...
eBook - ePubElasticity
Tensor, Dyadic, and Engineering Approaches
- Pei Chi Chou, Nicholas J. Pagano(Authors)
- 2013(Publication Date)
- Dover Publications(Publisher)
...In addition, the gradient of A is a vector which is normal to the surface A(x, y, z) = const. at each point on the surface. The divergence of a vector function is a scalar and is defined by (11.18) The curl of a vector function is expressed by (11.19) Another useful operator is (11.20) which is the Laplace operator. 11.3. Dyadic Notation In the symbolic representation, vector notation is not sufficient to treat second-order or higher-order tensor quantities, such as stress and strain. In this section we shall introduce a symbol, called a dyadic symbol, to represent a second-order tensor quantity. To treat third-order tensors, triadics must be introduced; and for fourth-order tensors, tetradics are needed. In this text, we shall limit our discussion in the symbolic approach to dyadics. It will be shown that all of the governing elasticity equations in Table 4.2 (or Table 10.1) can be represented by vector and dyadic notations. Generalized Hooke’s law, Eq. (10.6), which contains a fourth-order tensor quantity, cannot be written in dyadic notation. Dyadic symbols represent second-order tensor quantities, just as vector symbols are for first-order tensor quantities. In this text, a dyadic symbol is indicated by a letter in capital script. (In handwritten work one may use two bars over a letter to designate a dyadic symbol.) Thus, the stress tensor may be represented by the stress dyadic symbol. At this point, it is appropriate to review some of our discussion relating to vector quantities. A quantity with magnitude and direction and which adds according to the parallelogram law, or, a quantity whose components transform according to Eq. (8.12), is defined as a vector quantity, or simply, a vector. A vector quantity is also called a first-order tensor quantity, or simply, a first-order tensor. A vector quantity (or first-order tensor quantity) is represented in tensor notation by its three components, e.g., u i...
