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Arc Length
Multivariable Calculus - Lecture 2 -
Edward Frenkel
, UC Berkeley
Torus Homology
Flux in 3D
Curl
Removable Discontinuity
Mean Value Theorem
Finding Eigenvalues
Conway's Zip Proof
Classical Geometries
Equation of a Tangent Line
Visualizing Functions With Multiple Variables
Finding Determinants by Cofactors
Right-hand Rule
Integrals of Trigonometric Functions
Scale Factor
Scalar Multiplication of Linear Transformation
Convergence Tests for Improper Integrals
Impulse Inputs
Distance to a Plane
Derivatives
Universal Covering Space
Graphing With No Critical Points
Washer Method
Power Rule
Changing Coordinate Systems for Integrals
3D Determinants
Matrix Form of a System
Set Notation for Real Numbers
Surface Area in Spherical Coordinates
Traces
Critical Points in Non-linear Autonomous Systems
Degenerate Matrix Eigenvectors
Surface Integrals Over Scalar Functions
Example Graphs in Polar Coordinates
Group Structure of a Circle
Finding the Pseudo-Inverse
Divergence Theorem Proof
Simplical Complexes
Checking for Continuity
Maxwell's Equation
Lifting Homotopy Lemma
Orthonormal Basis
Integral Bounds With Changed Variables
Completing the Square in Integration
Linear Independence
First Order ODEs
Finding Maximums And Minimums (Multivariable)
Linear Systems
Consequences of Borsuk-ulam
Applications of Dot Product
Divergence Intuition
Euler's Method
Five Color Theorem
Existence And Uniqueness Theorem
Solving Min/max With Constraints
Integrals with Absolute Value
Derivative of Secant
Geometric Proof of Mean Value Theorem
One-Sided Limits
Cover Up Method (Partial Fractions)
Advantages of Hyperbolic Geometry
Divergence Theorem
Acceleration Vector
Eigenvalues
Deltahedrons
Properties of Integrals
Intersections of Planes
First-Order Autonomous ODEs
One-dimensional Objects
Finding Maximums And Minimums
3D Vector Fields
Finding Minimums with Matrices
Antiderivatives
Green's Theorem in Conservative Fields
Polar Representation of Complex Numbers
Non-independent Variables
Homotopic Paths
Limit Comparison
Lagrange Multipliers
2-Oriented Graphs
Dot Product
Small World Graphs
Derivative of Natural Log
Simply-Connected Regions
Homomorphism of Fundamental Group
Wronskians of Two Solutions
Surface Integrals Over Vector Fields
Desargues' Theorem
Cramer's Rule
Surface Area of a Sphere
Complete Eigenvalues
Inverse Matrix
Squeeze Theorem
Parametrization
Compatible Orientations
Euler's Method Step Size
Proof of Green's Theorem
Solving Systems With Elimination
Non-orientable Surfaces
Eigenvalues of a Reflection Matrix
Finding Volume Using the Determinant
Rolle's Theorem
Drawing Direction Fields
Image of a Transformation
Area in Polar Coordinates
Euler's Formula
Conservative Fields
Curl in 3D
Geometric Approach to Line Integrals
Path Independence
Rule of Sarrus
Gram-Schmidt Process
Symmetric Matrices
Tangent Lines of Surfaces
Critical Points (Multivariable)
Related Rates
Solid of Revolution
Inhomogeneous Systems
Fundamental Theorem of Algebra
Undoing Trig Substitutions
Parametric Equation of a Line
Vector Subspaces
Euler Characteristic And Cuffs
Green's Theorem
Links
Back Substitution
Kuratowski's Theorem
Finding Volume Using Vectors
Vector Fields
Component of a Vector Along Another Vector
Manipulating Limits
3D Coordinate System
Parametric Tangent Lines
Computing Antiderivatives
Riemann Surface
Matricies
Permutations (Linear Algebra)
Jump Discontinuity
Limit Cycles
Retraction
Basis of Column Space
Integral Test for Convergence
Volume of Sphere Portions
Matrices to Solve Equations
Torus as Orbit Space
Polar Coordinates
Properties of Stereographic Projections
LU-Decomposition
Simplex Orientation
Markov Matrix Eigenvalues
Cross-Caps and Handles
Brouwer Fixed-Point Theorem
Combinatorial Surfaces
RK2
Cross Product
Integration With Trig Substitution
Distance Between Two Planes
Area Between Curves with Horizontal Slices
Rowspace
Vector Projection
Elimination Failure
Gauss-Jordan Elimination
Center of Mass
Finding Area Using Vectors
L'hopital's Rule with Zero to the Zero Power
Solving Linear ODEs With Laplace Transforms
Euclidean Geometry
Fourier Matrix
Global vs. Local Extremes
Integration By Parts
General Formula for Determinants of Square Matrices
Green's Theorem For Flux
Line Integrals in 3D
Equivalence Relation
Saddle Points
Continuum
Knots
Vector Length
Weighted Averages
N-Holed Torus
Simpson's Rule
Scaling Variables
Girard's Theorem
Vector Triple Product Expansion
Torus as Configuration Space
Dual Polygons
Antipodes
Spherical Triangle Area
Permutation Matrix
Variation of Parameters
Projective Plane
Reflections in Spherical Geometry
Equations of Tangent Planes
Equations of Planes
Exponential Least Squares Interpolation
Matrix-Vector Multiplication
Runge-Katta Method
Linear ODEs
Knot Diagrams
Positive Definite Matrix Test
Optimization with Lagrange Multipliers
Stokes' Theorem
Position Vector
Critical Points
Graphs (Linear Algebra)
Dot Product Properties
Optimization Problems
Finding Eigenvectors
Flux
Calculating Volume With Integrals
Laplace Transform Existence Test
Determinants
Reflection Transformation
Work in a Vector Field
Orientability
Orthogonal Complement
Parallel or Perpendicular to Plane
Sun of Linear Transformations
Geometric Series
Dimension (of a vector space)
Mobius Band as Moduli Space
Solving Differential Equations with Matrices
Eigenvalue Stability
Cycle Definition
Change of Basis
Finding Determinants by Elimination
Graphing Using Derivatives
Column Space
Matrix of Second Derivatives
Limit Cycles in Quadratic Systems
Order of Integration in Cylindrical Coordinates
Decoupling Linear Systems
Platonic Solids
Basis
Spherical Coordinates
Markov Matrix
Preimages
Linear Combinations
Fundamental Theorem of Calculus
Finding Normalized Solutions
Right Hand vs. Left Hand Riemann Sums
Wronskians
Determining Invertibility of a Matrix
Linear Transformation Matrix
Changing Variable in Double Integrals
Substitutions in ODEs
Unit Tangent Vector
Second Order Linear ODEs
Graph of Arccos
Quadratic Surfaces
Circles in Moduli Space
Indefinite Integrals
Wavelets Basis
Harvesting
Projections onto Subspaces
Spheres
Quotients (Homology)
Vector Spaces
Derivative of Sine
Laplace Transform
Inflection Point
Comparing Integrals of Two Curves
Derivative of Arctangent
Convolution
Cylindrical Coordinates
Arc Length
Unit Vectors
Solving The Inhomogeneous Equation
Formal Definition of a Limit
Klein Bottles
Binomial Theorem
Riemann Sphere
Applications of Cross Product
Six Color Theorem
Directional Derivatives
FInding Extreme Values
Checking for Differentiability
Complex Eigenvalues
Differentials
Constant Multiple Rule
Orthogonal Complement of Null Space
Applications of Flux
Newton's Method Failure
Reidermeister Moves
Horizontal Asymptotes
Superposition Principle
Derivative of Trigonometric Functions
Limits
Surjective Functions (onto)
Integrating Factor
Eigenvalues of a Rotation Matrix
Eigenvalues of a Projection Matrix
Positive Definite Matrix
Symmetric Matrix Eigenvectors
Chain Rule (Multivariable)
Angle Sum Identity
Derivatives of Exponentials
Chain Rule
Gradient Fields
Superposition Principle Proof
Covering Spaces
Stokes' Theorem Vs. Green's Theorem
Second Partial Derivatives
Right Inverses
Advantages of Polar Form
Homeomorphism
Correctness of Euler's Method
Singular Value Decomposition (SVD)
Orthogonal Subspaces
Lagrange Multipliers for Unbounded Curves
Level Curves and Extreme Values
Torus
Second Fundamental Theorem of Calculus Proof
Four Fundamental Subspaces
Repeated Eigenvalues
Definite Integrals
Nullspace
Exponential Input Theorem
Vertical And Horizontal Tangent Lines
Real And Equal Roots
Graphs (Topology)
Calculating Probability with Integrals
Second Derivatives
Eigenvalues of a Triangular Matrix
Product Rule
Integrals of Odd Functions
Line Integrals
Reduced Row Echelon Form (RREF)
Matrix Rotation
Area of Region Bounded by a Curve
A-Transpose x A Invertibility
Disk Method
Direct Vs. Inverse Substitutions
Fundamental Matrix
Linear Transformations
Complex Methods
Cofactor (Linear Algebra)
Second Fundamental Theorem of Calculus
Numerical Methods of Solving ODEs
Simplex Standard Form
Transition Points
Covering Spaces And Fundamental Groups
Dimension of Null Space
Two-Sided Inverses
Non-orientability
Polygonal Representations of Circles
Jordan Blocks
Exponential Shift Rule
Taylor Expansion
Riemann Sums
Column Picture of a System
Block Matrix Multiplication
Length of Complex Vectors
Derivative of Cosine
Motion Represented in Parametric Equations
Laplace Transforms of Derivatives
Inverse of Square Matrix
Elimination Matrix
Pappus's Hexagon Theorem
Perpendicular Eigenvalues
Improper Integrals
Topology Introduction
Real And Distinct Roots
Exponential Shift in Laplace Transform
Second Derivative Test (Multivariable)
Calculating Speed From The Velocity Vector
Pick's Theorem
Two-Sided Limits
Row Space and Null Space Orthogonality
Velocity Vector
Similar Matrices
Direction Fields
Derivative as Linear Transformation
Smoothing a Piece-wise Function
Cylinders
Injective Functions
Orthonormalising Vectors
Two-Dimensional Surfaces
Torque
Improper Integrals of Second Kind
Logarithmic Differentiation
Resonance
Euler's Method Equations
Eigenvector Basis
Converting from Polar to Rectangular Coordinates
Determinants of Transposes
Uniqueness of an Antiderivative
Partial Fraction Decomposition
Infinite Series
Torus Knots
Pascal's Theorem
Optimization Problems (Multivariable)
Determining if Line And Plane Intersect
Derivatives of Inverse Functions
Homology
Determining Diagoalizability
Polytope Vertex Curvature
Covering Spaces of Torus
Gradient Field Test
Polytope Curvature
Surface Integrals
Convolution Formula Proof
Diffusion Equation
Characteristic Equation of a System
Calculating the Distortion Factor
Derivatives of Polynomials
Undamped Oscillation
Linear Spanning
Archimedean Solids
Homogenous ODEs
Lifting Path Lemma
Surface With Boundary
Non-linear Autonomous Systems
Genus
Existence of Limit Cycles
Vector-Valued Functions
Invertible Change of Basis Matrix
Stability Conditions
Hermitian Transpose
Newton's Method
Time-Independent ODEs
Measuring Connected Components
Determining Position With Parametric Equation
Hyperbolic Trig Substitution
Test
Sphere as Projective Line Over Complex Numbers
Derivative of Sine Intuition
Integrating Parametric Curves
Eigenvalues of a Linear Transfrom
Adding Vectors
Geometric Interpretation of Derivatives
Heun's Method
Gradient Vectors
Definite Integral With Substitution
Determining Orthogonality
Knot Invariants
Fourier Series
Vectors
Concavity
Inversive Plane
Cauchy-Schwarz Inequality
Average Value With Respect to Arclength
Least Squares Interpolation
Surface Area of a Revolution
Eigenspaces
Affine Plane
First Derivative Test
Non-Differentiable Functions
Linearization
Applications of Differentials
Damped Resonance
Gauss-green Theorem
Area Between Curves
Singular Matrix
U Substitution
Turning Numbers of Smooth Curves
Classification of Combinatorial Surfaces
Parallelogram Rule (vectors)
Derivative of Tangent
Linear Dependence
Orthogonal Vectors
Second Derivative Test
Winding Numbers
Fourier Basis
Planar Graphs
Dimension of Circle
Calculating Instantaneous Speed
Projections onto Orthonormal Bases
Conversion to First-order Equation
Damped Oscillation
Fundamental Theorem of Calculus For Line Integrals
Partial Derivatives
Matrix Multiplication
Normal Vectors
Ambient Space
Moment of Inertia
Linear Space
Critically Damped
Solving Systems With Matrices
Quotient Rule
Origins of Positive Definite Matrices
Scaling Vectors
Fubini's Theorem
Power Series
Implicit Differentiation
Finding The Angle Between Vectors
Laplace Transforms With Discontinuities
Divergence Theorem Interpretation
Linear Homogeneous Systems
Linear Approximation vs Mean Value Theorem
Finding Global Extremes
Partial Fraction Decomposition With Improper Fractions
Quadratic Approximation
Fast Fourier Transform
Jordan Form
Identity Matrix
Double Integrals (Multivariable)
Complex Roots
Differential Equations
Applications of Convolution
Finding The Potential
Periodic Definite Integral
Monotonicity Theorem
Simplicies
Separation of Variables
Ham Sandwich Theorem (Stone-Tukey)
Convergent Power Series
Transposes
Complex Vectors
Continuity
Triple Integrals
Finding Coefficients of a Fourier Series
Canonical Form
Total Number of Platonic Solids
Vector Transformation
Eigenvectors
Turn-Angles
Linear Approximation
Divergence
Jacobian
Computing Nullspace
Platonic Solids in Higher Dimensions
Double Integrals in Polar Coordinates
Hypervolume
L'hopital's Rule Failure
Integration Using The Half-Angle Formula
Borsuk-ulam Theorem
Fundamental Group
Left Inverses
Contour Plots
Dirac Delta Function
Inhomogeneous OEDs
Euler Number of a Combinatorial Surface
Rotation Transformation
Projection Matrix
Implicit vs Explicit Functions
Diagonal Matrix
Mobius Band
Parametric Form
Ratio Test
Reduction Formulae
Graphing Derivatives
Least Squares Regression
Finding Tangent Planes
Area Under a Curve
Finding Average Value in a Region
Infinite Discontinuity
Convergence Tests for Infinite Series
Average Value of a Curve
Shell Method
Geometric View of Differential Equations
Integrating Over a Region
Level Curves
Total Differential
Incidence Matrix
Oscillations
Properties of Gradients
Adding Matrices
Taylor's Rule
Row Picture of a System
Systems of Differential Equations
Equations of Lines
Chains(Topology)
Alexander-Conway Polynomial
Applications of Projections
Polyhedra
Linear Subspace
Flux Through a Surface
Trapezoid Rule
Path Multiplication
L'hopital's Rule
Zero Vector
Applications of Winding Numbers
Tangent Plane Approximation
Triangle Inequality
Derivative of Vector Function
Types of Knots
Hyperbolic Geometry
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Parametric Tangent Lines
Vertical And Horizontal Tangent Lines
Integrating Parametric Curves
Equation of a Tangent Line
Polar Coordinates
Surface Area of a Sphere
Parametric Form
Surface Area of a Revolution
Velocity Vector
Calculating Speed From The Velocity Vector
Position Vector
Acceleration Vector
Unit Tangent Vector
Example Graphs in Polar Coordinates