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The fundamental theorem of calculus links differentiation and integration, showing that these two operations are essentially inverses of each other. It has two parts: the first states that an antiderivative of a function is the integral of the function over an interval, and the second states that the integral of a function is the change of its antiderivative over an interval.
Learn why division by zero is a problematic special case in mathematics and computing. Explore different interpretations, definitions, and number systems of division by zero, and their implications for limits, functions, and errors.
Euler's formula is a mathematical formula that relates the trigonometric functions and the complex exponential function. It has many applications in complex analysis, physics, chemistry, and engineering, and can be proved using different methods such as differentiation, power series, or polar coordinates.
Learn the definition, history and applications of the limit of a function in calculus and analysis. Find out how to use the (ε, δ)-definition of limit for functions of a single variable or several variables.
Euler's identity is the equality e^1 = -1, where e is Euler's number and i is the imaginary unit. It is a special case of Euler's formula and a famous example of mathematical beauty, as it connects five fundamental constants and shows profound symmetry.
The chain rule is a formula that expresses the derivative of the composition of two differentiable functions in terms of the derivatives of the functions. Learn the intuitive explanation, history, statement, and examples of the chain rule and its applications to composites of more than two functions, quotient rule, and inverse functions.
The gradient theorem states that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. The web page explains the theorem, its proof, its applications, and its converse.
Learn the definitions and properties of vector calculus operators such as gradient, divergence, curl, and Laplacian. Find the formulas and examples of various identities involving these operators and their derivatives.