Vectors
The name for single real numbers to contrast from Vectors, an ordered group of real numbers
What is Scaler?
Second order derivative of a path [ex. z(t)] or first order derivative of the Velocity function
What is acceleration?
Also found in Single Variable Calculus this method includes using the sum of rectangular prisms to approximate the volume under a surface
What is Riemann Sums?
To find the average of this (say t) find the scaler integral of t over the scaler integral of 1
What is (Average) Temperature?
Denoted as ∇ f, is the collection of all its partial derivatives into a vector.
What is Gradient?
A geometric method of visualizing Vector Addition using Arrows as Vectors
What is the “head-to-tail” method?
The length L(x) of a C1 path x: [a, b] → Rn is found by integrating this
What is Speed/Magnitude of Velocity?
Triple integrating 1 gives you Volume while triple integrating 3D shape W of density p(x,y,z) gives this
What is Mass?
The vector integral of F (force field) is this on a particle along path c
What is work?
The path that is in the direction parallel to the gradient or represented as ∇ f(a, b) / ||∇ f(a, b)||
What is path of steepest ascent?
Denoted a × b, an operation in three-dimensional space that results in a new vector which is perpendicular (orthogonal) to both original vectors
What is Cross Product?
The operator used in Divergence, Curl, and Gradients
What is the del operator (∇)?
Spherical coordinates utilize these 3 greek letters instead of X,Y, and Z in Cartesian Coordinates
What is ϕ (Phi), θ (Theta), ρ (Rho)?
According to Green's Theorem the region bounded by vector c (Region D) must be to vector c's left or in other words this
What is positively oriented?
A point that is both a minimum on one axis and a maximum on another
What is a Saddle Point?
Coordinate system that utilizes a plane filled with concentric circles centered at the origin and rays emanating from the origin. Denoted as (θ,r)
What is Polar coordinates?
A matrix of all first-order partial derivatives of a vector-valued function
What is Jacobian Matrix?
Fubini's Theorem states under certain assumptions the double integral over a rectangle can be calculated by using iterated integrals and this certain quality does not matter
What is Order of Integration?
To show that vector field F is conservative compute when vector F = ∇ f (gradient of f) or this
What is Curl(Vector F)?
This greek letter also called the Lagrange multiplier is used to represent a scaler value in a method for solving constrained optimization problems
What is λ (lambda)?
The process of representing a geometric object by expressing its coordinates as functions of one or more independent variables
Ex:(x = 5 − t, y = 3, and z = 7t + 8)
What is parameterization?
Divergence turns a Vector field into this type of field
What is Scalar Field?
(DAILY DOUBLE)
Nationality of Guido Fubini
What is Italian?
Gauss' Theorem also known by this name takes the triple integral of ∇· F(Vector field)
What is Divergence Theorem?
Name the Theorem: A function is on a closed, bounded interval then f must have both a global maximum and a global minimum somewhere on X.
What is Extreme Value Theorem?