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A metric tensor field g on M assigns to each point p of M a metric tensor g p in the tangent space at p in a way that varies smoothly with p. More precisely, given any open subset U of manifold M and any (smooth) vector fields X and Y on U , the real function g ( X , Y ) ( p ) = g p ( X p , Y p ) {\displaystyle g(X,Y)(p)=g_{p}(X_{p},Y_{p})} is ...
Vector and tensor calculus in general curvilinear coordinates is used in tensor analysis on four-dimensional curvilinear manifolds in general relativity, [8] in the mechanics of curved shells, [6] in examining the invariance properties of Maxwell's equations which has been of interest in metamaterials [9] [10] and in many other fields.
the point's direction from the pole relative to the direction of the polar axis, a ray drawn from the pole. The distance from the pole is called the radial coordinate, radial distance or simply radius, and the angle is called the angular coordinate, polar angle, or azimuth. [1] The pole is analogous to the origin in a Cartesian coordinate system.
Vector and tensor calculus in general curvilinear coordinates is used in tensor analysis on four-dimensional curvilinear manifolds in general relativity, [11] in the mechanics of curved shells, [9] in examining the invariance properties of Maxwell's equations which has been of interest in metamaterials [12] [13] and in many other fields.
Once the radius is fixed, the three coordinates (r, θ, φ), known as a 3-tuple, provide a coordinate system on a sphere, typically called the spherical polar coordinates. The plane passing through the origin and perpendicular to the polar axis (where the polar angle is a right angle) is called the reference plane (sometimes fundamental plane).
In this polar decomposition, the unit circle has been replaced by the line x = 1, the polar angle by the slope y/x, and the radius x is negative in the left half-plane. If x 2 ≠ y 2 , then the unit hyperbola x 2 − y 2 = 1 and its conjugate x 2 − y 2 = −1 can be used to form a polar decomposition based on the branch of the unit hyperbola ...
On a Riemannian manifold, a normal coordinate system at p facilitates the introduction of a system of spherical coordinates, known as polar coordinates. These are the coordinates on M obtained by introducing the standard spherical coordinate system on the Euclidean space T p M .
The second expression is for a polar graph = parameterized by =. Now let C ( t ) = ( r ( t ) , θ ( t ) , ϕ ( t ) ) {\displaystyle \mathbf {C} (t)=(r(t),\theta (t),\phi (t))} be a curve expressed in spherical coordinates where θ {\displaystyle \theta } is the polar angle measured from the positive z {\displaystyle z} -axis and ϕ ...