Joseph-Louis Lagrange was born on January 25, 1736 in Turin, Sardinia-Piedmont (which is now known as Italy). He studied at the College of Turin where his favorite subject was classic Latin. After reading Halleys 1693 work on the use of algebra in optics Lagrange became very interested in mathematics and astronomy. Unfortunately for Lagrange he did not have the benefit of studying with the leading mathematicians, so he became self-motivated and was self-taught.
Then in 1754 he got the opportunity to publish his first mathematical work, which was an analogy between the binomial theorem and the successive derivatives of the product functions. Lagrange sent some of his works to Euler and impressed him greatly. Euler was so overcome that by his work that he appointed Lagrange professor of mathematics at the Royal Artillery School in Turin. Then in 1756 he was elected to the Berlin Academy. This then led Lagrange being a founding member of what would eventually become the Royal Academy of Science in Turin.
In 1766, Lagrange accepted Eulers position as the director of the Berlin Academy. While director of the academy Lagrange produced some of his greatest work. In 1772 he shared a prize with Euler on the three body problems. Two years later he won a prize on the motion of the moon, and then in 1780 he won a prize on perturbations of the orbits of comets by the planets. Lagrange was made a member of the committee of the Academic des Sciences to standardize weights and measures in 1790. They worked on the metric system and supported a decimal base.
In 1808 Lagrange was named to the Legion of Honour and Count of the Empire by Napoleon. Lagrange later died in 1813. Lagrange, along with Euler and Bernoullis, developed the calculus of variations for dealing with mechanics. He was responsible for laying the groundwork for a different way of writing down Newtons Equation of Motion. This is called Lagrangian Mechanics. It accomplishes the same thing that Newtons Equations provided but the form of the equations is actually better because the form does not change when the coordinate system used to describe a problem changes.
He also introduced a new variable for studying conservative systems, which is now called the Lagrangian. However, rewriting Newtons Equations did nothing fundamentally different it was simply a more convenient way to write it down and solve it. In the Calculus Early Transcendentals Tenth Edition Lagrange multipliers is used to solve max-min problems. In 1755 Lagrange developed the Lagrange multipliers to solve maximum and minimum problems in geometry. Today this method is important to economics, engineering, and mathematics.
The method states that the extreme values of a function f(x,y,z) whose variables are subject to a constraint g(x,y,z) = 0 are to be found on the surface g = 0 at the points where for some scalar (called a Lagrange multiplier). For example, suppose that f(x,y,z) and g(x,y,z) are differentiable. To find the local maximum and minimum values of f subject to the constraint g(x,y,z) = 0 find the values of x,y,z and simultaneously satisfy the equations
For the functions of two independent variables the appropriate equations are In his first edition of Mecanique Analytique he defined the volume and the surface area by where the equation of the surface is given by z=f(x,y) and dz=Pdx +Qdy. He did not give enough explanations here but he noted that the double integral signs indicated that the two integrations must be performed successively. It was later in 1811, in the second edition of his Mecanique Analytique that he introduced the general notion of a surface integral.