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In uniform scaling with a non-zero scale factor, all non-zero vectors retain their direction (as seen from the origin), or all have the direction reversed, depending on the sign of the scaling factor. In non-uniform scaling only the vectors that belong to an eigenspace will retain their direction. A vector that is the sum of two or more non ...
In mathematics, the discrete exterior calculus (DEC) is the extension of the exterior calculus to discrete spaces including graphs, finite element meshes, and lately also general polygonal meshes [1] (non-flat and non-convex). DEC methods have proved to be very powerful in improving and analyzing finite element methods: for instance, DEC-based ...
The geometric interpretation of Newton's method is that at each iteration, it amounts to the fitting of a parabola to the graph of () at the trial value , having the same slope and curvature as the graph at that point, and then proceeding to the maximum or minimum of that parabola (in higher dimensions, this may also be a saddle point), see below.
In calculus, the trapezoidal rule (also known as the trapezoid rule or trapezium rule) [a] is a technique for numerical integration, i.e., approximating the definite integral: (). The trapezoidal rule works by approximating the region under the graph of the function f ( x ) {\displaystyle f(x)} as a trapezoid and calculating its area.
In mathematics, holomorphic functional calculus is functional calculus with holomorphic functions. That is to say, given a holomorphic function f of a complex argument z and an operator T , the aim is to construct an operator, f ( T ), which naturally extends the function f from complex argument to operator argument.
The six independent scalar products g ij =h i.h j of the natural basis vectors generalize the three scale factors defined above for orthogonal coordinates. The nine g ij are the components of the metric tensor, which has only three non zero components in orthogonal coordinates: g 11 =h 1 h 1, g 22 =h 2 h 2, g 33 =h 3 h 3.
Scaling (geometry) a similar notion in vector spaces Homothetic center , the center of a homothetic transformation taking one of a pair of shapes into the other The Hadwiger conjecture on the number of strictly smaller homothetic copies of a convex body that may be needed to cover it
Scale analysis is very useful and widely used tool for solving problems in the area of heat transfer and fluid mechanics, pressure-driven wall jet, separating flows behind backward-facing steps, jet diffusion flames, study of linear and non-linear dynamics. Scale analysis is an effective shortcut for obtaining approximate solutions to equations ...