The lagrange function, commonly denoted L is the lagrangian of a system. Usually it is the kinetic energy - potential energy (in the case of a particle in a conservative potential).

The lagrange equation is the equation that converts a given lagrangian into a system of equations of motion. It is d/dt(\partial L/\partial qdot)-\partial L/\partial q.

Partial differential equations are mathematical equations that involve two or more independent variables, an unknown function, and partial derivatives of the unknown function. Even the explanation is confusing! If, however, anyone chooses to learn about PDE there are classes offered at any institution of higher learning.

Differentials can be used to approximate a nonlinear function as a linear function.

They can be used as a "factory" to quickly find partial derivatives.

They can be used to test if a function is smooth.

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