Universal Law of Gravitation:
The Universal Law of Gravitation, formulated by Sir Isaac Newton, states that every particle of matter attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. In simple terms, every object in the universe attracts every other object with a force that depends on their masses and the distance between them.

Mathematical Expression:
The mathematical expression for the Universal Law of Gravitation is as follows:
F = G * (m1 * m2) / r^2

Where:
- F represents the gravitational force between two objects.
- G is the gravitational constant, which is approximately equal to 6.674 × 10^-11 Nm^2/kg^2.
- m1 and m2 are the masses of the two objects.
- r is the distance between the centers of the two objects.

Importance:
The Universal Law of Gravitation holds great significance in the field of physics and our understanding of the universe. Here are some key points explaining its importance:

1. Planetary Motion: The law explains the motion of planets around the Sun and other celestial bodies. It helps in understanding the stability of planetary orbits and predicting their paths accurately.

2. Kepler's Laws: The Universal Law of Gravitation is the foundation for Kepler's laws of planetary motion. These laws describe the motion of planets and satellites, providing insights into their elliptical orbits and varying speeds.

3. Gravitational Force Calculation: The law enables the calculation of the gravitational force between any two objects, regardless of their size or distance. This is crucial in various fields, such as astrophysics, engineering, and space exploration.

4. Gravitational Field: The law defines the concept of a gravitational field, which allows us to understand how objects influence the space around them. It helps us analyze the behavior of objects in the presence of strong gravitational forces.

5. Understanding Tides: The Universal Law of Gravitation explains the phenomenon of tides on Earth. The gravitational pull of the Moon and the Sun on Earth's oceans creates tidal effects, which are crucial for navigation, coastal ecosystems, and other marine activities.

6. Gravitational Waves: The law also forms the basis for the study of gravitational waves. These waves, predicted by Einstein's theory of general relativity, provide valuable information about massive cosmic events like black hole mergers.

In conclusion, the Universal Law of Gravitation is a fundamental principle in physics that describes the gravitational interactions between objects. Its mathematical expression allows for precise calculations, and its applications extend to various fields, contributing to our understanding of the universe and enabling technological advancements.