Explain divergence in the context of a series

A series diverges if the sum does not approach a finite number as the number of terms increases.

Define arc length

The length of a curve defined by a function.

Define a derivative at a point

The slope of the tangent line to a function at that point.

Define the concept of a gradient field

A vector field representing the gradient of a scalar function.

Define a vector field

An assignment of a vector to each point in a subset of space.

Define convergence in the context of a series

A series converges if the sum approaches a finite number as the number of terms goes to infinity.

Define a partial derivative

The derivative of a multivariable function with respect to one variable while keeping others constant.

Define definite integral

The integral of a function over a specified interval.

Define a Riemann Sum

An approximation of the integral of a function using finite sums.

Define a conservative vector field

A vector field where the line integral from one point to another is independent of the path.

Define a differential

An infinitesimally small change in a function, related to the derivative.

Define a Laplace transform

An integral transform that converts a function of a real variable into a function of a complex variable.

Define a PDE (Partial Differential Equation)

An equation involving partial derivatives of a function of several variables.

Define the Mean Value Theorem

There exists at least one point where the function's derivative equals the function's average rate of change over an interval.

Define the ratio test for series convergence

A test that uses the ratio of successive terms to determine a series' convergence.

Define limit

The value that a function approaches as the input approaches some value.

Define curl in vector analysis

A measure of the rotation of a vector field.

Define Euler's Method

A numerical method to approximate solutions of differential equations.

Define a surface integral

The integral of a scalar field over a surface.

Define implicit differentiation

Finding the derivative of a function defined by an equation.

Define the concept of a divergence of a vector field

A scalar representing the magnitude of a source or sink at a given point in a vector field.

Define a Maclaurin Series

A Taylor Series centered at zero.

Define a parametric equation

An equation where the variables are expressed as functions of a parameter.

Define the concept of optimization

The process of finding the maximum or minimum values of a function.

Define the disk method for volume calculation

A method to find the volume of a solid of revolution using disks.