What is the equation for a sphere?
The general equation of the sphere is x2 + y2 + z2 = r2 and in this article, we will learn about deriving the equation of a sphere along with its volume and surface area.
What formula do we use to calculate the magnitude of a vector expressed in Cartesian coordinates?
Using the Pythagorean Theorem, we can obtain an expression for the magnitude of a vector in terms of its components. Given a vector a=(a1,a2), the vector is the hypotenuse of a right triangle whose legs are length a1 and a2. Hence, the length of the vector a is ∥a∥=√a21+a22.
How do you find the vector of a sphere?
The general vector equation of a sphere is r2−2r. c+(c2−a2)=0, where ∣a∣ is radius and c is the centre of sphere.
Which is the vector function for the sphere?
One vector expression for the sphere is ⟨ 1 − v 2 cos u, v ⟩ —this emphasizes the tube structure, as it is naturally viewed as drawing a circle of radius 1 − v 2 around the z -axis at height v . We could also take a cue from spherical coordinates, and write ⟨ sin
How to convert Cartesian coordinates to spherical coordinates?
ρ2 = x2 + y2 + z2 Converting points from Cartesian or cylindrical coordinates into spherical coordinates is usually done with the same conversion formulas. To see how this is done let’s work an example of each. Example 1 Perform each of the following conversions.
How are vector fields defined in spherical coordinates?
Vectors are defined in spherical coordinates by (r, θ, φ), where r is the length of the vector, θ is the angle between the positive Z-axis and the vector in question (0 ≤ θ ≤ π), and φ is the angle between the projection of the vector onto the X-Y-plane and the positive X-axis (0 ≤ φ < 2π).
How to find the cylindrical coordinates of a surface?
Likewise, if we have a point in Cartesian coordinates the cylindrical coordinates can be found by using the following conversions. Let’s take a quick look at some surfaces in cylindrical coordinates. Example 1 Identify the surface for each of the following equations. In two dimensions we know that this is a circle of radius 5.