To define some physical quantities, there is a need to specify both magnitude as well as direction. For example, when the U.S. Coast Guard dispatches a …
Vectors are usually described in terms of their components in a coordinate system. Even in everyday life, we naturally invoke the concept of orthogonal …
The Cartesian coordinate system is a very convenient tool to use when describing the displacements and velocities of objects and the forces acting on …
Spherical coordinate systems are preferred over Cartesian, polar, or cylindrical coordinates for systems with spherical symmetry. For example, to describe …
Vectors can be multiplied by scalars, added to other vectors, or subtracted from other vectors. The vector sum of two (or more) vectors is called the …
It is cumbersome to find the magnitudes of vectors using the parallelogram rule or using the graphical method to perform mathematical operations like …
The scalar multiplication of two vectors is known as the scalar or dot product. As the name indicates, the scalar product of two vectors results in a …
Vector multiplication of two vectors yields a vector product, with the magnitude equal to the product of the individual vectors multiplied by the sine of …
Two vectors can be multiplied using a scalar product or a vector product. The resultant of a scalar product is scalar, while with vector products, the …
In mathematics and physics, the gradient and del operator are fundamental concepts used to describe the behavior of functions and fields in space. The …
The divergence of a vector field at a point is the net outward flow of the flux out of a small volume through a closed surface enclosing the volume, as …
The first order operators using the del operator include the gradient, divergence and curl. Certain combinations of first order operators on a scalar or …
A line integral for a vector field is defined as the integral of the dot product of a vector function with an infinitesimal displacement vector along a …
Multi-dimensional and transient flows play a key role in many areas of science, engineering, and health sciences but are often not well understood. The …
