Generalized Coordinates: Vector Geometry
In what ways are Cartesian coordinates different from the "generalized" spherical and cylindrical coordinates? When do I have to take care?
QUESTIONS
(Score is number right minus number wrong.)
Cartesian, spherical and cylindrical unit vectors are constants. Equations (2a), (2b), and (2c) are correct.
True
False
Cartesian unit vectors are constants, but cylindrical unit vectors are not necessarily constants. Equations (3a) and (3b) are correct.
True
False
Cartesian unit vectors are constants both in length and direction, but cylindrical unit vectors are not necessarily constant in direction for different field points.
True
False
The position of an arbitrary field point is given in CARTESIAN coordinates by
Equation (4a)
Equation (4b)
Equation (4c)
Equation (4d)
Equation (4e)
Equation (4f)
Equation (4g)
Equation (4h)
Equation (4i)
The position of an arbitrary field point is given in SPHERICAL coordinates by
Equation (4a)
Equation (4b)
Equation (4c)
Equation (4d)
Equation (4e)
Equation (4f)
Equation (4g)
Equation (4h)
Equation (4i)
Equation (4e) has no angular dependence (theta or phi) as a function of position.
True
False
Equation (4e) does have implicit angular dependence (theta or phi) as a function of position. The r-unit vector depends on theta and phi. See Equation (1).
True
False
The position of an arbitrary field point is given in CYLINDRICAL coordinates by
Equation (4a)
Equation (4b)
Equation (4c)
Equation (4d)
Equation (4e)
Equation (4f)
Equation (4g)
Equation (4h)
Equation (4i)
For a plane surface, the direction of a differential vector area is ambiguous in algebraic sign and must be determined by some additional convention or consideration.
True
False
A differential area at the bottom of the cylinder in the figure (x,y plane, z=0) that is directed outward from the volume of the cylinder is given by
Equation (5a).
Equation (5b).
A differential vector area on the surface of a SPHERE and directed OUTWARD is given by
Equation (5a)
Equation (5b)
Equation (5c)
Equation (5d)
Equation (5e)
Equation (5f)
Equation (5g)
Equation (5h)
Equation (5i)
Equation (5j)
A differential vector area on the curved surface of a CYLINDER and directed INWARD is given by
Equation (5a)
Equation (5b)
Equation (5c)
Equation (5d)
Equation (5e)
Equation (5f)
Equation (5g)
Equation (5h)
Equation (5i)
Equation (5j)
A differential vector area in the phi = 45 degree plane of a CYLINDER and directed COUNTERCLOCKWISE as seen from above is given by
Equation (5a)
Equation (5b)
Equation (5c)
Equation (5d)
Equation (5e)
Equation (5f)
Equation (5g)
Equation (5h)
Equation (5i)
Equation (5j)
A completely arbitrary differential vector step taken away from a given field point is given in Cartesian coordinates as Equation (6a). In SPHERICAL coordinates, the step is given by
Equation (6a)
Equation (6b)
Equation (6c)
Equation (6d)
Equation (6e)
Equation (6f)
Equation (6g)
Equation (6h)
Equation (6i)
A completely arbitrary differential vector step taken away from a given field point is given in Cartesian coordinates as Equation (6a). In SPHERICAL coordinates, the LENGTH of the step is given by
Equation (7a)
Equation (7b)
Equation (7c)
Equation (7d)
Equation (7e)
Equation (7f)
Equation (7g)
Equation (7h)
Equation (7i)
A completely arbitrary differential vector step taken away from a given field point is given in Cartesian coordinates as Equation (6a). In CYLINDRICAL coordinates, the step is given by
Equation (6a)
Equation (6b)
Equation (6c)
Equation (6d)
Equation (6e)
Equation (6f)
Equation (6g)
Equation (6h)
Equation (6i)
A completely arbitrary differential vector step taken away from a given field point is given in Cartesian coordinates as Equation (6a). In CYLINDRICAL coordinates, the LENGTH of the step is given by
Equation (7a)
Equation (7b)
Equation (7c)
Equation (7d)
Equation (7e)
Equation (7f)
Equation (7g)
Equation (7h)
Equation (7i)
A completely arbitrary differential vector step taken away from a given field point is given in Cartesian coordinates as Equation (6a). If the step is CONSTRAINED to lie in a plane parallel to the x-y plane (z=constant), the LENGTH of the step is given by
Equation (7a)
Equation (7b)
Equation (7c)
Equation (7d)
Equation (7e)
Equation (7f)
Equation (7g)
Equation (7h)
Equation (7i)
A completely arbitrary differential vector step taken away from a given field point is given in Cartesian coordinates as Equation (6a). If the step is CONSTRAINED to lie in a plane parallel to the x-y plane (z=constant) and the step is further CONSTRAINED to lie along some path determined by Equation (8a), the LENGTH of the step is given by
Equation (9a)
Equation (9b)
Equation (9c)
Equation (9d)
A completely arbitrary differential vector step taken away from a given field point is given in Cartesian coordinates as Equation (6a). If the step is CONSTRAINED to lie in a plane parallel to the x-y plane (z=constant) and the step is further CONSTRAINED to lie along the parabola determined by Equation (8b), the LENGTH of the step is given by Equation (10a)
True
False
If the step is CONSTRAINED to lie on the surface of a sphere and the step is further CONSTRAINED to lie along the helix determined by Equation (8c), the LENGTH of the step is given by Equation (10b)
True
False
EQUATIONS
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