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What is the expression for the metric tensor in spherical coordinates?
The expression for the metric tensor in spherical coordinates is given by: gij beginpmatrix 1 0 0 0 r2 0 0 0 r2 sin2(theta) endpmatrix
How can the rotation matrix be expressed in terms of spherical coordinates?
The rotation matrix can be expressed in terms of spherical coordinates by using the azimuthal angle (), the polar angle (), and the radial distance (r) to determine the orientation of the rotation.
How is the derivation of unit vectors in spherical coordinates carried out?
In spherical coordinates, unit vectors are derived by taking the partial derivatives of the position vector with respect to the spherical coordinates (r, , ) and normalizing them to have a magnitude of 1. This process involves using trigonometric functions and the chain rule to find the components of the unit vectors in the radial, azimuthal, and polar directions.
What is the expression for the unit vector r hat in spherical coordinates?
The expression for the unit vector r hat in spherical coordinates is r hat sin(theta)cos(phi) i sin(theta)sin(phi) j cos(theta) k.
What are the spherical to cartesian unit vectors?
The spherical to cartesian unit vectors are used to convert coordinates between spherical and cartesian systems. They are denoted as ( hatr ), ( hattheta ), and ( hatphi ), representing the radial, azimuthal, and polar directions respectively.
What is the expression for the metric tensor in spherical coordinates?
The expression for the metric tensor in spherical coordinates is given by: gij beginpmatrix 1 0 0 0 r2 0 0 0 r2 sin2(theta) endpmatrix
How do you derive navier stokes equation in spherical coordinates?
To derive the Navier-Stokes equations in spherical coordinates, we start with the general form of the Navier-Stokes equations in Cartesian coordinates and apply the transformation rules for spherical coordinates ((r, \theta, \phi)). This involves expressing the velocity field, pressure, and viscous terms in terms of the spherical coordinate components. The continuity equation is also transformed accordingly to account for the divergence in spherical coordinates. Finally, we reorganize the resulting equations to isolate terms and ensure they reflect the physical properties of fluid motion in a spherical geometry.
How can the rotation matrix be expressed in terms of spherical coordinates?
The rotation matrix can be expressed in terms of spherical coordinates by using the azimuthal angle (), the polar angle (), and the radial distance (r) to determine the orientation of the rotation.
How is the derivation of unit vectors in spherical coordinates carried out?
In spherical coordinates, unit vectors are derived by taking the partial derivatives of the position vector with respect to the spherical coordinates (r, , ) and normalizing them to have a magnitude of 1. This process involves using trigonometric functions and the chain rule to find the components of the unit vectors in the radial, azimuthal, and polar directions.
How the heat conduction equations for spherical and cylindrical coordinates were derived?
The coordinates for equations dealing with cylindrical and spherical conduction are derived by factoring in the volume of the thickness of the cylindrical control. Coordinates are placed into a Cartesian model containing 3 axis points, x, y, and z.
What is the expression for the unit vector r hat in spherical coordinates?
The expression for the unit vector r hat in spherical coordinates is r hat sin(theta)cos(phi) i sin(theta)sin(phi) j cos(theta) k.
What are the spherical to cartesian unit vectors?
The spherical to cartesian unit vectors are used to convert coordinates between spherical and cartesian systems. They are denoted as ( hatr ), ( hattheta ), and ( hatphi ), representing the radial, azimuthal, and polar directions respectively.
What has the author Ivan Istvan Mueller written?
Ivan Istvan Mueller has written: 'Introduction to surveying' -- subject(s): Surveying 'Spherical and practical astronomy' -- subject(s): Spherical astronomy 'Gravimetic and celestial geodesy' -- subject(s): Dictionaries, Geodesy 'Reference coordinate systems' -- subject(s): Coordinates, Geodetic surveys, Inertial reference systems, Spherical coordinates, Standards
What are curvillinear coordinates?
Curvilinear coordinate systems are a means of describing an object/point's position based on angle and distance from the origin. For example, polar coordinates are Curvilinear coordinates for R2 (2D space). Spherical coordinates are Curvilinear coordinates for R3 (3D space) If you need to know more: http://en.wikipedia.org/wiki/Curvilinear_coordinates
What are the coordinates are 40 N 120 W?
Using the spherical coordinate system, they would be (6378 km, 40, -120).
What is the expression for kinetic energy in spherical coordinates?
The expression for kinetic energy in spherical coordinates is given by: KE 0.5 m (r2) ('2 sin2() '2) where KE is the kinetic energy, m is the mass of the object, r is the radial distance, is the polar angle, is the azimuthal angle, and ' and ' are the angular velocities in the respective directions.
How do you solve for a spherical triangle?
If by sperical triangle you mean a triangle on the surface of a sphere, you will need 3 dimensional coordinate geometry. Whether you use polar coordinates or linear coordinates will depend on what you want to "solve".
