So far, so good. For a better description, what is the radius of the hemisphere, and where is its center (i.e., the center of the sphere that the hemisphere is half of)?yango_17 said:Homework Statement
Sketch the solid whose spherical coordinates (ρ, φ, θ):
0≤ρ≤1, 0≤φ≤(pi/2)
Homework Equations
The Attempt at a Solution
I was thinking that since ρ represented the distance from the point of the origin and φ represented the angle between the positive z-axis and the ray through the origin and any point, that the surface represented by such an inequality would be something like the top hemisphere of a 3d sphere. Any feedback would be appreciated.
The center is at a particular point on the z-axis. Care to guess which one?yango_17 said:The radius of the hemisphere would be 1, and the center would be on the z-axis. I'm not entirely sure what the absence of θ signifies.
Spherical coordinate inequalities are mathematical expressions that define a region in three-dimensional space using spherical coordinates. They consist of three inequalities, each involving the three spherical coordinates: radius, inclination angle, and azimuthal angle.
To sketch a solid given spherical coordinate inequalities, you first need to understand the inequalities and how they restrict the values of the spherical coordinates. Then, you can plot the corresponding points and connect them to form the boundaries of the solid. It may be helpful to use a graphing calculator or software to visualize the solid.
Spherical coordinate inequalities are based on spherical coordinates, which use a radius and two angles to locate a point in three-dimensional space. Cartesian inequalities, on the other hand, are based on Cartesian coordinates, which use three perpendicular axes to locate a point. The main difference is in the coordinate systems used, but both can be used to define regions in three-dimensional space.
No, spherical coordinate inequalities can also be used to define other types of regions in three-dimensional space, such as surfaces or volumes. They are commonly used in physics and engineering to describe the limitations or boundaries of a system.
Spherical coordinate inequalities can be advantageous in certain situations because they can simplify the representation of a region in three-dimensional space. For example, if a region has spherical symmetry, it may be easier to define using spherical coordinate inequalities rather than Cartesian inequalities. Spherical coordinates can also be useful for certain types of calculations, such as finding the volume of a sphere or cone.