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A graph of the vector-valued function r(z) = 2 cos z, 4 sin z, z indicating a range of solutions and the vector when evaluated near z = 19.5. A common example of a vector-valued function is one that depends on a single real parameter t, often representing time, producing a vector v(t) as the result.
It is expedient to use vector notation: let = (,,), let be a parameter, let () be the parametric representation of a curve , and let ˙ be its tangent vector. The optical length of the curve is given by A [ C ] = ∫ t 0 t 1 n ( X ) X ˙ ⋅ X ˙ d t . {\displaystyle A[C]=\int _{t_{0}}^{t_{1}}n(X){\sqrt {{\dot {X}}\cdot {\dot {X}}}}\,dt.}
More generally, for a function of n variables (, …,), also called a scalar field, the gradient is the vector field: = (, …,) = + + where (=,,...,) are mutually orthogonal unit vectors. As the name implies, the gradient is proportional to, and points in the direction of, the function's most rapid (positive) change.
In detail, if h is a displacement vector represented by a column matrix, the matrix product J(x) ⋅ h is another displacement vector, that is the best linear approximation of the change of f in a neighborhood of x, if f(x) is differentiable at x. [a] This means that the function that maps y to f(x) + J(x) ⋅ (y – x) is the best linear ...
In kinematics, objects' paths through space are commonly described as parametric curves, with each spatial coordinate depending explicitly on an independent parameter (usually time). Used in this way, the set of parametric equations for the object's coordinates collectively constitute a vector-valued function for position.
Concretely, on R 3 this is given by: 1-forms and 1-vector fields: the 1-form a x dx + a y dy + a z dz corresponds to the vector field (a x, a y, a z). 1-forms and 2-forms: one replaces dx by the dual quantity dy ∧ dz (i.e., omit dx), and likewise, taking care of orientation: dy corresponds to dz ∧ dx = −dx ∧ dz, and dz corresponds to dx ...
If a curve γ represents the path of a particle, then the instantaneous velocity of the particle at a given point P is expressed by a vector, called the tangent vector to the curve at P. Mathematically, given a parametrized C 1 curve γ = γ(t), for every value t = t 0 of the parameter, the vector ′ = | = is the tangent vector at the point P ...
A formula for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the cubic equation 4x 3 − 3x + d = 0, where is the value of the cosine function at the one-third angle and d is the known value of the cosine function at the full angle.