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In MATLAB, the function kron (A, B) is used for this product. These often generalize to multi-dimensional arguments, and more than two arguments. In the Python library NumPy, the outer product can be computed with function np.outer(). [8] In contrast, np.kron results in a flat array.
A pole figure is a graphical representation of the orientation of objects in space. For example, pole figures in the form of stereographic projections are used to represent the orientation distribution of crystallographic lattice planes in crystallography and texture analysis in materials science .
MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages. Although MATLAB is intended primarily for numeric computing, an optional toolbox uses the MuPAD symbolic engine allowing access to symbolic computing abilities.
An example of a multivector-valued function of multivectors that is linear but is not an outermorphism is grade projection where the grade is nonzero, for example projection onto grade 1: x ∧ y 1 = 0 {\displaystyle \langle x\wedge y\rangle _{1}=0}
For example, a G-set is equivalent to a functor from G to the category of sets, Set, and a linear representation is equivalent to a functor to the category of vector spaces over a field, Vect K. Given two representations, ρ and σ, of G in C, an equivariant map between those representations is simply a natural transformation from ρ to σ.
The definition of the exterior algebra can be extended for spaces built from vector spaces, such as vector fields and functions whose domain is a vector space. Moreover, the field of scalars may be any field. More generally, the exterior algebra can be defined for modules over a commutative ring.
For example, in order to find the optimal airfoil shape for an aircraft wing, an engineer simulates the airflow around the wing for different shape variables (e.g., length, curvature, material, etc.). For many real-world problems, however, a single simulation can take many minutes, hours, or even days to complete.
The field of numerical analysis predates the invention of modern computers by many centuries. Linear interpolation was already in use more than 2000 years ago. Many great mathematicians of the past were preoccupied by numerical analysis, [5] as is obvious from the names of important algorithms like Newton's method, Lagrange interpolation polynomial, Gaussian elimination, or Euler's method.