Mass
In everyday usage, mass and "weight" are often used interchangeably. For instance, a person's weight may be stated as 75 kg. In a constant gravitational field, the weight of an object is proportional to its mass, and it is unproblematic to use the same unit for both concepts. But because of slight differences in the strength of the Earth's gravitational field at different places, the distinction becomes important for measurements with a precision better than a few percent, and for places far from the surface of the Earth, such as in space or on other planets. Conceptually, "mass" (measured in kilograms) refers to an intrinsic property of an object, whereas "weight" (measured in newtons) measures an object's resistance to deviating from its natural course of free fall, which can be influenced by the nearby gravitational field. No matter how strong the gravitational field, objects in free fall are weightless, though they still have mass.
Linear density
Consider a long, thin rod of mass formula_1 and length formula_2. To calculate the average linear mass density, formula_3, of this one dimensional object, we can simply divide the total mass, formula_1, by the total length, formula_2:
If we describe the rod as having a varying mass (one that varies as a function of position along the length of the rod, formula_7), we can write:
Each infinitesimal unit of mass, formula_9, is equal to the product of its linear mass density, formula_10, and the infinitesimal unit of length, formula_11:
The linear mass density can then be understood as the derivative of the mass function with respect to the one dimension of the rod (the position along its length formula_13)