Questions related to understanding line integrals, vector fields, surface integrals, the theorems of Gauss, Green and Stokes. Some related tags are (multivariable-calculus) and (differential-geometry).

Vector analysis, a branch of mathematics that deals with quantities that have both magnitude and direction.

Some physical quantities like the mass or the temperature at some point only have magnitude. We can represent these quantities by number alone (with the appropriate units) and so we call them scalars. There are other physical quantities that have magnitude and direction, called vector. Their magnitude can stretch or shrink, and their direction can reverse. These quantities can be added in such a way that takes into account both direction and magnitude.

Force is an example of a quantity that acts in a certain direction with some magnitude that we measure in Newtons. When two forces act on an object, the sum of the forces depends on both the direction and magnitude of either one. Position, displacement, velocity, acceleration, force, momentum, and torque are all physical quantities that can be mathematically represented by vectors.

One of the most difficult problems in understanding physics is learning how to represent these physical quantities as mathematical vectors.

Vector calculus was developed from quaternion analysis by J. Willard Gibbs and Oliver Heaviside near the end of the $~19{th}~$ century, and most of the notation and terminology was established by Gibbs and Edwin Bidwell Wilson in their $~1901~$ book, Vector Analysis.

In the conventional form using cross products, vector calculus does not generalize to higher dimensions, while the alternative approach of geometric algebra, which uses exterior products does generalize, as discussed below.

Applications:

When we apply vectors to physical quantities it’s nice to keep in the back of our minds all these formal properties. However from the physicist’s point of view, we are interested in representing physical quantities like displacement, velocity, acceleration, force, impulse, momentum, torque, and angular momentum as vectors. We can’t add force to velocity or subtract momentum from torque. We must always understand the physical context for the vector quantity. So instead of approaching vectors as formal mathematical objects we shall instead consider the following essential properties that enable us to represent physical quantities as vectors.

References:

https://en.wikipedia.org/wiki/Vector_calculus

http://web.mit.edu/8.01t/www/materials/modules/ReviewA.pdf

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