Chapter 2
Vectors and Scalars
![Introduction to Scalars](https://cloudfront.jove.com/files/media/science-education/science-education-thumbs/12608.jpg)
Many familiar physical quantities can be specified completely by giving a single number and the appropriate unit. For example, "a class period…
![Introduction to Vectors](https://cloudfront.jove.com/files/media/science-education/science-education-thumbs/12609.jpg)
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…
![Polar and Cylindrical Coordinates](https://cloudfront.jove.com/files/media/science-education/science-education-thumbs/12611.jpg)
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 Coordinates](https://cloudfront.jove.com/files/media/science-education/science-education-thumbs/14691.jpg)
Spherical coordinate systems are preferred over Cartesian, polar, or cylindrical coordinates for systems with spherical symmetry. For example, to…
![Vector Algebra: Method of Components](https://cloudfront.jove.com/files/media/science-education/science-education-thumbs/12613.jpg)
It is cumbersome to find the magnitudes of vectors using the parallelogram rule or using the graphical method to perform mathematical operations like…
![Scalar Product (Dot Product)](https://cloudfront.jove.com/files/media/science-education/science-education-thumbs/12614.jpg)
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…
![Scalar and Vector Triple Products](https://cloudfront.jove.com/files/media/science-education/science-education-thumbs/14688.jpg)
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,…
![Gradient and Del Operator](https://cloudfront.jove.com/files/media/science-education/science-education-thumbs/14689.jpg)
In mathematics and physics, the gradient and del operator are fundamental concepts used to describe the behavior of functions and fields in space.…
![Second Derivatives and Laplace Operator](https://cloudfront.jove.com/files/media/science-education/science-education-thumbs/14692.jpg)
The first order operators using the del operator include the gradient, divergence and curl. Certain combinations of first order operators on a scalar…
![Line, Surface, and Volume Integrals](https://cloudfront.jove.com/files/media/science-education/science-education-thumbs/14693.jpg)
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…
![High-speed Particle Image Velocimetry Near Surfaces](https://cloudfront.jove.com/files/thumbs/50559_t.png)
High-speed Particle Image Velocimetry Near Surfaces
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Multi-dimensional and transient flows play a key role in many areas of science, engineering, and health sciences but are often not well understood.…
![Echo Particle Image Velocimetry](https://cloudfront.jove.com/files/thumbs/4265_t.png)
The transport of mass, momentum, and energy in fluid flows is ultimately determined by spatiotemporal distributions of the fluid velocity field.1…