91Ó°ÊÓ

A sphere is a perfectly round three-dimensional shape, similar to a ball. It can be defined mathematically as the set of all points in space that are equidistant from a given point called the center. The distance from any point on the sphere to the center is known as the radius.
The standard equation for a sphere centered at the origin with radius \( R \) is given by:

• \( x^2 + y^2 + z^2 = R^2 \)

This means any point \( (x, y, z) \) that satisfies this equation lies on the sphere's surface.
Spheres are often encountered in physics, engineering, and various mathematical contexts, such as when studying geometric properties and volume.
In the given exercise, the problem confirms whether a specific parametric curve lies on the surface of a sphere by reformulating the parametric equations to fit the equation of a sphere.

In mathematics, a curve is a continuous and smooth flowing line without sharp corners or singularities. Curves can exist in two-dimensional or three-dimensional spaces and are described using parametric equations, which define a set of parameters often related to time.
The given exercise describes a curve using a parametric equation in three-dimensional space:

• \( \mathbf{r}(t) = \langle a\sin(mt)\cos(nt), b\sin(mt)\sin(nt), c\cos(mt) \rangle \)

Curves can take various shapes and must be analyzed to understand their behavior, such as knowing if they lie on certain geometrical surfaces like spheres.
The properties of a curve can be explored by algebraically manipulating its equations, which in this case involves trigonometric identities and the condition \( a^2 + b^2 = c^2 \) to verify if it lies on a sphere.

Trigonometric identities are mathematical equations involving trigonometric functions that hold true for all values of the variables involved. They simplify complex trigonometric expressions and are crucial for solving problems in calculus and geometry.
Common trigonometric identities include:

• \( \sin^2(\theta) + \cos^2(\theta) = 1 \)
• \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \)
• \( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \)

In the provided exercise, the identity \( \sin^2(mt) + \cos^2(mt) = 1 \) is particularly important. It helps confirm that the sum of squares \( x^2 + y^2 + z^2 \) conforms to an equation of a sphere when manipulated appropriately.
Understanding and applying these identities helps reduce the complexity of trigonometric expressions, making it easier to draw conclusions about the nature of curves and their relation to geometric shapes.

Vectors are mathematical objects used to represent quantities that have both magnitude and direction. They are essential in physics, engineering, and mathematics, particularly when analyzing motion and forces in three-dimensional space.
A vector can be denoted in terms of its components along the coordinate axes. In 3D, a vector \( \mathbf{v} \) is written as:

• \( \mathbf{v} = \langle v_x, v_y, v_z \rangle \)

In the given problem, the equation \( \mathbf{r}(t) = \langle a \sin(mt)\cos(nt), b \sin(mt)\sin(nt), c \cos(mt) \rangle \) is a parametric representation of a vector that describes a point on the curve as a function of the parameter \( t \).
Manipulating vectors and understanding their relationships is key to solving problems that involve trajectories, rotations, and positioning in space. The exercise demonstrates how vectors are used to describe geometrical shapes like spheres through parametric equations.