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Vector Calculus
Published in Ahmad Shahid Khan, Saurabh Kumar Mukerji, Electromagnetic Fields, 2020
Ahmad Shahid Khan, Saurabh Kumar Mukerji
The dot product of vector operator ∇ and any vector (say F) gives a scalar function and is called the divergence of that vector, provided that the vector operator ∇ operates on the vector F. It is written as ∇• F or divF. In Equation 3.9b, it was noted that dot product obeys the commutative law (i.e. A·B=B·A). This however is not true here (i.e. F·∇≠∇·F) as ∇ is not a simple vector but a differential vector operator. Note that F • ∇ is a scalar differential operator, while ∇• F is a scalar mathematical expression.
Transport equations and steady-state flow
Published in Ian Acworth, Investigating Groundwater, 2019
In this context we can understand the divergence as the difference between the mass inflow rate and the mass outflow rate per unit volume. Another useful vector operator is the Laplacian ∇2. This can be written as: ∇2()=∂2()∂x2+∂2()∂y2+∂2()∂z2
Vector and Tensor Calculus
Published in Tasos C. Papanastasiou, Georgios C. Georgiou, Andreas N. Alexandrou, ViscousFluid Flow, 2021
Tasos C. Papanastasiou, Georgios C. Georgiou, Andreas N. Alexandrou
The nabla operator is a vector operator which acts on scalar, vector, or tensor fields. The result of its action is another field, the order of which depends on the type of the operation. In the following, we will first define the various operations of ∇ in Cartesian coordinates, and then discuss their forms in curvilinear coordinates.
Image registration for varicose ulcer classification using KNN classifier
Published in International Journal of Computers and Applications, 2018
R. R. Bhavani, G. Wiselin Jiji
The gradient is a vector operator denoted by δ (referred to as “del”) which, when applied to a function f, represents its directional derivatives. For example, consider a two-dimensional function f (x, y) which shows elevation above sea level at points x and y. The gradient f (x, y) or f is the vector pointing in the direction of the steepest slope at that point. The gradient is calculated by:
Method for detecting damage in carbon-fibre reinforced plastic-steel structures based on eddy current pulsed thermography
Published in Nondestructive Testing and Evaluation, 2018
Xuan Li, Zhiping Liu, Xiaoli Jiang, Gabrol Lodewijks
where ki is a scalar term. The divergence characterizes the diffusion (or convergence) rate of the fluid volume of an object. In vector calculus, divergence is a vector operator that measures the magnitude of a vector field’s source or sink at a given point. For example,