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VectorCalculus[VectorSpace] - create a tangent vector space at a given point in a given coordinate system
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Calling Sequence
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VectorSpace(c, origin)
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Parameters
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c
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(optional) symbol or symbol[name, name, ...]; specify the coordinate system, possibly indexed by the coordinate names
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origin
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Vector or list; free or position Vector or list; specify the root point of the tangent vector space
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Description
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The VectorSpace(c, origin) command returns a module representing the tangent vector space rooted at point origin in c coordinates.
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The order of the arguments may be reversed. If no coordinate system argument is present, the current coordinate system is used.
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The root point origin can be specified as a free or position Vector or as a list of coordinates. If it is a free or position Vector, the coordinate system attribute is checked and conversion done accordingly. If it is a list, the components are expected to be in c coordinates.
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Once coordinates c are specified, the unit basis vectors are evaluated at the point origin. If any of the unit basis vectors evaluate to zero an error is raised.
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The returned module is a valid right hand side for the space attribute on Vectors. If this attribute is present, it qualifies the Vector as a rooted Vector, and determines its root point and coordinate system.
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The module itself has 3 exports:
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GetRootPoint
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Return the root point of the vector space
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GetCoordinates
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Return the coordinate system of the vector space
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Vector(comps)
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Create a rooted Vector based on the vector space, with components comps. comps is expected to be of type list(algebraic).
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Examples
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![vs := VectorSpace(spherical[r, phi, theta], [1, (1/2)*Pi, (1/2)*Pi])](/support/helpjp/helpview.aspx?si=4285/file03412/math135.png)
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![Matrix([[`Type: `, `Vector Space`], [`Coordinates: `, spherical[r, phi, theta]], [`Root Point: `, [1, (1/2)*Pi, (1/2)*Pi]]])](/support/helpjp/helpview.aspx?si=4285/file03412/math145.png)
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![v := vs:-Vector([0, 1, 0])](/support/helpjp/helpview.aspx?si=4285/file03412/math149.png)
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](/support/helpjp/helpview.aspx?si=4285/file03412/math152.png)
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![Matrix([[`Type: `, `Rooted Vector`], [`Components: `, [0, 1, 0]], [`Coordinates: `, spherical[r, phi, theta]], [`Root Point: `, [1, (1/2)*Pi, (1/2)*Pi]]])](/support/helpjp/helpview.aspx?si=4285/file03412/math159.png)
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](/support/helpjp/helpview.aspx?si=4285/file03412/math166.png)
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![v := VectorSpace(polar[r, theta], `<,>`(x, y)):-Vector([a, b])](/support/helpjp/helpview.aspx?si=4285/file03412/math170.png)
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![Matrix([[`Type: `, `Rooted Vector`], [`Components: `, [a, b]], [`Coordinates: `, polar[r, theta]], [`Root Point: `, [(x^2+y^2)^(1/2), arctan(y, x)]]])](/support/helpjp/helpview.aspx?si=4285/file03412/math177.png)
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![Matrix([[`Type: `, `Vector Space`], [`Coordinates: `, polar[r, theta]], [`Root Point: `, [(x^2+y^2)^(1/2), arctan(y, x)]]])](/support/helpjp/helpview.aspx?si=4285/file03412/math184.png)
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