Calculus III Memorization & Recognition Guide

Unit Circle

A circle of radius 1 centered at the origin in the coordinate plane, used to define trigonometric functions.

Pythagorean Identity

The fundamental identity in trigonometry stating that sin²t + cos²t = 1.

Tangent of an angle

The ratio of the sine to the cosine of the angle, expressed as tan(t) = sin(t)/cos(t).

Sphere Equation

The equation of a sphere in three-dimensional space is given by x² + y² + z² = R², where R is the radius.

Ellipsoid Equation

The equation of an ellipsoid in space is given by \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1.

Hyperbolic Paraboloid

A saddle-shaped surface defined by the equation z = x² - y².

Gradient of a function

A vector consisting of the partial derivatives of the function with respect to each variable, pointing in the direction of steepest ascent.

Direction of Maximum Increase

The direction in which the function increases most rapidly, given by the normalized gradient vector.

Jacobian Matrix

A matrix of all first-order partial derivatives of a vector-valued function.

Cross Product of Two Vectors

A vector operation that results in a vector that is perpendicular to the plane formed by the two input vectors.

Implicit Differentiation

A method to find the derivative of a dependent variable defined implicitly by an equation involving both variables.

Arc Length in 2D

Calculated as (L = \int_a^b\sqrt{(x'(t))^2 + (y'(t))^2}\,dt) over the interval from a to b.

Partial Derivative

The derivative of a function with respect to one variable while holding the other variables constant.

Distance Formula in 3D

The distance between two points in 3D space is given by d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}.

Tangent Plane

The plane that best approximates a surface at a given point, expressed in the form z = f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b).

Chain Rule for Functions of Several Variables

Used for finding the derivative of a composition of functions with multiple parameters.

Trigonometric Identities

Equations that relate the angles and sides in right triangles; they are used to simplify and solve trigonometric equations.

Unit Vector

A vector with a magnitude of one, often used to indicate direction.

Quadrant Signs for Trigonometric Functions

Determines the positivity or negativity of sine, cosine, and tangent in each of the four quadrants of the Cartesian plane.

Cylinders in 3D

Surfaces created by extending a two-dimensional shape along a third dimension, represented by equations with one variable missing.

Volume of a Sphere

The amount of space within a sphere, calculated using the formula (V = \frac{4}{3} \pi R^3), where R is the radius.

Level Curves

Curves along which a function of two variables takes a constant value, represented by the equation f(x,y) = c.

Normal Vector

A vector that is perpendicular to a surface at a given point.

Helix

A three-dimensional spiral, often represented parametrically by (\mathbf r(t) = \langle a\cos(t), a\sin(t), bt \rangle).