Advanced Calculus and Differential Equations

An equation that involves an unknown function and its derivatives, often used to describe physical phenomena.

An equation that describes the motion of a particle based on Newton's second law, where the force acting on the particle is a function of time and position.

An equation that describes the rate of decay of a radioactive substance, where the rate of decay is proportional to the concentration of the substance.

An equation that describes the propagation of waves in a medium, such as sound waves in air.

A series is said to be divergent if its sequence of partial sums does not converge to a finite limit.

A series is said to be convergent if its sequence of partial sums converges to a finite limit.

The least upper bound (LUB) of a set of numbers is the smallest number that is greater than or equal to all numbers in the set.

A periodic function repeats its values at regular intervals.

The period of a periodic function is the length of one complete cycle of the function.

Fourier series representation of an arbitrary function in terms of sines and cosines.

A series is said to be absolutely convergent if the sum of the absolute values of its terms converges.

A series is said to be conditionally convergent if it converges, but the sum of the absolute values of its terms diverges.

A point where a function's partial derivatives are zero, but the function does not have a maximum or minimum value.

Conditions that must be met for a function to possibly have a maximum or minimum value at a given point. This means that the partial derivatives of the function must be zero at that point.

Conditions that determine whether a function actually has a maximum, minimum, or neither at a point where the necessary conditions are satisfied.

A function that can be expressed as a sum of terms, each of which is a power of the variables, where the powers of all variables in each term add up to the same value.

A vector field where the curl is zero.

The value of the partial derivative of a function with respect to x, evaluated at a specific point.

The limit of a function as the variables approach a given point does not exist.

A function is continuous at a point if the function's value at that point is equal to the limit of the function as the variables approach that point.

A mathematical equation that describes the relationship between a function and its derivatives with respect to multiple independent variables.

The order of the highest derivative in a differential equation. For example, a second-order differential equation contains a second derivative.

A function that maps one variable to another (e.g., f(x) = x² maps each input x to its square).

Describes the variation of temperature in a solid material over time and space.

A differential equation where the function depends on a single independent variable (e.g., dy/dt = ky describes exponential growth).

A differential equation where the function f(t, y) is linear in its second argument y, meaning it can be written as f(t, y) = a(t)y + b(t).

A function that maps a set of input variables to a single output (e.g., f(x, y, z) = x + y + z maps three inputs to their sum).

A point in two-dimensional space is represented as an ordered pair (x, y) where x and y are real numbers. The set of all such points forms the two-dimensional Euclidean space, denoted by R².

A point in three-dimensional space is represented as an ordered triple (x, y, z) where x, y, and z are real numbers. The set of all such points forms the three-dimensional Euclidean space, denoted by R³.

Functions of two variables have the form z = f(x, y) where x and y are the independent variables and z is the dependent variable. For example, f(x, y) = x² + y² is a function of two variables.

The limit of a function f(x, y) as (x, y) approaches (a, b) is denoted by lim_(x,y)→(a,b) f(x,y) = l. It means that for every 𝛿 > 0, there exists a 𝛿>0 such that |f(x, y) - l| < 𝛿 whenever 0 < (x - a)² + (y - b)² < 𝛿².

An implicit function is defined by an equation of the form f(x, y) = C, where C is a constant. In this case, both x and y are considered functions of x. Differentiating both sides with respect to x, we get ∂f/∂x + ∂f/∂y(dy/dx) = 0, which can be solved to find dy/dx.

The Jacobian of u and v with respect to x and y, denoted by J, is defined as ∂(u,v)/∂(x,y) = ∂u/∂x * ∂v/∂y - ∂u/∂y * ∂v/∂x. It measures how much u and v change with respect to changes in x and y.

The Jacobian of (x, y) with respect to (u, v) is the reciprocal of the Jacobian of (u, v) with respect to (x, y). This means JJ' = 1.

If u and v are functions of r and θ, then the Jacobian of (u, v) with respect to (x, y) can be calculated as ∂(u,v)/∂(x,y) = ∂(u,v)/∂(r,θ) * ∂(r,θ)/∂(x,y). This relates Jacobians in different coordinate systems.

A method for solving differential equations by multiplying both sides by an integrating factor, which is a function that makes the left-hand side a total derivative.

A differential equation where the coefficients of the derivatives are only constants.

A function of the form ce^(at), where c and a are constants, that is a solution to the linear differential equation y'(t) = ay(t) + b.

A type of differential equation where the dependent variable and its derivatives appear only in the first power and are not multiplied together. The coefficients may be functions of the independent variable.

A technique used to solve ordinary differential equations by multiplying both sides of the equation by a suitable function that makes the left-hand side a total derivative.

A differential equation where the highest order derivative is the second derivative.

A method used to solve differential equations by finding a particular solution that satisfies the initial or boundary conditions and then adding the general solution of the homogeneous equation.

A differential equation where the coefficients of the derivatives are functions of the independent variable.