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  2. Vector fields in cylindrical and spherical coordinates

    en.wikipedia.org/wiki/Vector_fields_in...

    ρ is the length of the vector projected onto the xy-plane, φ is the angle between the projection of the vector onto the xy-plane (i.e. ρ) and the positive x-axis (0 ≤ φ < 2π), z is the regular z-coordinate. (ρ, φ, z) is given in Cartesian coordinates by:

  3. Vector notation - Wikipedia

    en.wikipedia.org/wiki/Vector_notation

    Using the algebraic properties of subtraction and division, along with scalar multiplication, it is also possible to “subtract” two vectors and “divide” a vector by a scalar. Vector subtraction is performed by adding the scalar multiple of −1 with the second vector operand to the first vector operand. This can be represented by the ...

  4. Scalar (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Scalar_(mathematics)

    A scalar is an element of a field which is used to define a vector space.In linear algebra, real numbers or generally elements of a field are called scalars and relate to vectors in an associated vector space through the operation of scalar multiplication (defined in the vector space), in which a vector can be multiplied by a scalar in the defined way to produce another vector.

  5. Conversion between quaternions and Euler angles - Wikipedia

    en.wikipedia.org/wiki/Conversion_between...

    Let us define scalar and vector such that quaternion = (,). Note that the canonical way to rotate a three-dimensional vector v → {\displaystyle {\vec {v}}} by a quaternion q {\displaystyle q} defining an Euler rotation is via the formula

  6. Linear form - Wikipedia

    en.wikipedia.org/wiki/Linear_form

    If V is a vector space over a field k, the set of all linear functionals from V to k is itself a vector space over k with addition and scalar multiplication defined pointwise. This space is called the dual space of V, or sometimes the algebraic dual space, when a topological dual space is also considered.

  7. Covariance and contravariance of vectors - Wikipedia

    en.wikipedia.org/wiki/Covariance_and_contra...

    The components of a vector are often represented arranged in a column. By contrast, a covector has components that transform like the reference axes. It lives in the dual vector space, and represents a linear map from vectors to scalars. The dot product operator involving vectors is a good example of a covector.

  8. Spherical coordinate system - Wikipedia

    en.wikipedia.org/wiki/Spherical_coordinate_system

    The del operator in this system leads to the following expressions for the gradient and Laplacian for scalar fields, = ^ + ^ + ⁡ ^, = + ⁡ (⁡) + ⁡ = (+) + ⁡ (⁡) + ⁡ , And it leads to the following expressions for the divergence and curl of vector fields,

  9. Row and column vectors - Wikipedia

    en.wikipedia.org/wiki/Row_and_column_vectors

    The transpose (indicated by T) of any row vector is a column vector, and the transpose of any column vector is a row vector: […] = [] and [] = […]. The set of all row vectors with n entries in a given field (such as the real numbers ) forms an n -dimensional vector space ; similarly, the set of all column vectors with m entries forms an m ...