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Illustration of tangential and normal components of a vector to a surface. In mathematics, given a vector at a point on a curve, that vector can be decomposed uniquely as a sum of two vectors, one tangent to the curve, called the tangential component of the vector, and another one perpendicular to the curve, called the normal component of the vector.
The vector projection (also known as the vector component or vector resolution) of a vector a on (or onto) a nonzero vector b is the orthogonal projection of a onto a straight line parallel to b. The projection of a onto b is often written as proj b a {\displaystyle \operatorname {proj} _{\mathbf {b} }\mathbf {a} } or a ∥ b .
Illustration of tangential and normal components of a vector to a surface. The decomposition or resolution [16] of a vector into components is not unique, because it depends on the choice of the axes on which the vector is projected. Moreover, the use of Cartesian unit vectors such as ^, ^, ^ as a basis in which to represent a vector is not ...
A terminology often used in physics refers to the curl-free component of a vector field as the longitudinal component and the divergence-free component as the transverse component. [22] This terminology comes from the following construction: Compute the three-dimensional Fourier transform F ^ {\displaystyle {\hat {\mathbf {F} }}} of the vector ...
The curl of the vector field at any point is given by the rotation of an infinitesimal area in the xy-plane (for z-axis component of the curl), zx-plane (for y-axis component of the curl) and yz-plane (for x-axis component of the curl vector). This can be seen in the examples below.
Vector projection of a on b (a 1), and vector rejection of a from b (a 2). In mathematics, the scalar projection of a vector on (or onto) a vector , also known as the scalar resolute of in the direction of , is given by:
In such a presentation, the notions of length and angle are defined by means of the dot product. The length of a vector is defined as the square root of the dot product of the vector by itself, and the cosine of the (non oriented) angle between two vectors of length one is defined as their dot product. So the equivalence of the two definitions ...
Here, the upper index refers to the number of the coordinate or component, so x 2 refers to the second component, and not the quantity x squared. The index variable i is used to refer to an arbitrary component, such as x i. The divergence can then be written via the Voss-Weyl formula, [8] as: