Determine the surface for which the equation is given.&nbsp;p=sin⁡θsin⁡ϕ

Question

Determine the surface for which the equation is given.&amp;nbsp;p=sin⁡θsin⁡ϕ

ensojadasH · Accepted Answer

See the photo below:

Don Sumner · Accepted Answer

The given equation is p=sin(θ)sin(ϕ).This equation represents a mathematical surface in spherical coordinates. The variables θ and ϕ represent the spherical angles, and p represents the radial distance from the origin to a point on the surface.In spherical coordinates, θ represents the azimuthal angle (measured from the positive x-axis in the xy-plane), and ϕ represents the polar angle (measured from the positive z-axis).To visualize this surface, we can consider different values of θ and ϕ and calculate the corresponding values of p using the equation.For example, if we fix θ=π4 and vary ϕ from 0 to 2π, we can plot the resulting values of p. Similarly, we can fix ϕ=π4 and vary θ to obtain another plot.The resulting surface will depend on the range of θ and ϕ values chosen. The equation p=sin(θ)sin(ϕ) describes a surface that exhibits symmetry with respect to the xz-plane and the yz-plane. It is important to note that the range of θ and ϕ values should be appropriately chosen to obtain a meaningful representation of the surface.

nick1337 · Accepted Answer

Answer:(0,12,0)Explanation:To determine the surface for the given equation p=sin(θ)sin(ϕ), we can use spherical coordinates. In spherical coordinates, p represents the radial distance from the origin, θ represents the polar angle measured from the positive z-axis, and ϕ represents the azimuthal angle measured from the positive x-axis.To find the surface, we can express the equation in terms of spherical coordinates. Since p=sin(θ)sin(ϕ), we can substitute the spherical coordinate expressions into the equation:p=sin(θ)sin(ϕ)r=p·sin(θ)=sin(θ)sin(ϕ)·sin(θ)=sin2(θ)sin(ϕ)Using the relationship between Cartesian coordinates (x,y,z) and spherical coordinates (r,θ,ϕ), we have:x=r·sin(θ)·cos(ϕ)y=r·sin(θ)·sin(ϕ)z=r·cos(θ)Substituting the expression for r into the Cartesian coordinates, we get:x=sin2(θ)sin(ϕ)·sin(θ)·cos(ϕ)y=sin2(θ)sin(ϕ)·sin(θ)·sin(ϕ)z=sin2(θ)sin(ϕ)·cos(θ)Simplifying these equations, we obtain:x=sin3(θ)sin2(ϕ)·cos(ϕ)y=sin3(θ)sin2(ϕ)·sin(ϕ)z=sin3(θ)sin(ϕ)·cos(θ)Therefore, the surface described by the equation p=sin(θ)sin(ϕ) is given by the parametric equations:x=sin3(θ)sin2(ϕ)·cos(ϕ)y=sin3(θ)sin2(ϕ)·sin(ϕ)z=sin3(θ)sin(ϕ)·cos(θ)To find the point where x=0, y=12, and z=0, we substitute these values into the parametric equations:sin3(θ)sin2(ϕ)·cos(ϕ)=0sin3(θ)sin2(ϕ)·sin(ϕ)=12sin3(θ)sin(ϕ)·cos(θ)=0Solving these equations, we find that the point satisfying the given conditions is (0,12,0).

madeleinejames20 · Accepted Answer

To determine the surface for which the equation is given,

p = \sin θ \sin ϕ

, we can use spherical coordinates. In spherical coordinates,

p

represents the distance from the origin to a point,

θ

is the polar angle (measured from the positive

z

-axis), and

ϕ

is the azimuthal angle (measured from the positive

x

-axis in the

x y

-plane).
To express the equation

p = \sin θ \sin ϕ

in terms of spherical coordinates, we substitute the spherical coordinates for

p

,

θ

, and

ϕ

:

ρ = \sin θ \sin ϕ

Here,

ρ

represents the radial distance from the origin to a point.
This equation defines a surface in three-dimensional space. To visualize this surface, we can eliminate

ρ

and express the equation solely in terms of

θ

and

ϕ

:

\sin θ \sin ϕ = 0

Now, let's consider the possible values of

θ

and

ϕ

that satisfy this equation:
1. When

\sin θ = 0

and

\sin ϕ = 0

, we have

θ = k π

and

ϕ = m π

, where

k

and

m

are integers. In this case, the equation becomes

0 = 0

, which is true for all values of

θ

and

ϕ

. Therefore, the surface is not well-defined for this case.
2. When

\sin θ = 0

and

\sin ϕ \neq 0

, we have

θ = k π

and

ϕ \neq m π

. In this case, the equation becomes

0 = 0

, which is true for all values of

θ

and

ϕ

as long as

ϕ

is not an integer multiple of

π

. This implies that the surface contains all points on the

z

-axis except for the origin.
3. When

\sin θ \neq 0

and

\sin ϕ = 0

, we have

θ \neq k π

and

ϕ = m π

. In this case, the equation becomes

\sin θ = 0

, which is satisfied when

θ = k π

where

k

is an integer. This implies that the surface contains all points on the

x y

-plane.
In conclusion, the surface defined by the equation

p = \sin θ \sin ϕ

consists of all points on the

z

-axis (excluding the origin) and the

x y

-plane.

Identify the surface whose equation is given. p=\sin\theta\sin\phi

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