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Multivariable Calculus

Vectors

 Vectors. Definition, components, representation, examples, solved exercises, and Properties Vector Arithmetic. Dot product. Read more... 1545 words,πŸ•”8 minutes read, May 14, 2022.

Cross Products

 Vectors II. Determinant in space. Cross products. Properties. Solved exercises. Read more... 2712 words,πŸ•”13 minutes read, May 14, 2022.

Vectors II. The Dot Product.

 Vectors. The Dot Product. Solved exercises. Read more... 2144 words,πŸ•”11 minutes read, May 14, 2022.

Vectors III. Planes and Areas

 Calculating the Area of a Triangle Using Vectors Read more... 1545 words,πŸ•”8 minutes read, May 14, 2022.

Equations of Planes

 Find the Equation of a Plane Given Three Points. Vectors and the Geometry of Space. Solved exercises. Read more... 2219 words,πŸ•”11 minutes read, May 14, 2022.

Matrices: definitions, types, examples, and properties.

 Rotation Matrices. Inverse matrix. Properties. Trace of a matrix. Diagonal matrix. Lower and upper triangular matrix. Symmetric matrix. Read more... 3281 words,πŸ•”16 minutes read, May 14, 2022.

Systems of Linear Equations.

 Equations of planes. Solving Systems of Linear Equations. Exercises of vectors and planes. Read more... 3042 words,πŸ•”15 minutes read, May 14, 2022.

Parametric equations for lines and curves II

 Parametric equations for lines and curves. Graphing a Parametrically Defined Curve. A cycloid. Read more... 1822 words,πŸ•”9 minutes read, May 14, 2022.

Parametric equations for lines and curves.

 Parametric equations for lines and curves. Graphing a Parametrically Defined Curve. Eliminating the Parameter. Read more... 1980 words,πŸ•”10 minutes read, May 14, 2022.

Systems of Linear Equations II

 Equations of planes. Solving Systems of Linear Equations. Exercises of vectors and planes. Read more... 3627 words,πŸ•”18 minutes read, May 14, 2022.

Functions of two variables II

 Functions of two variables. Sketching graphs. Read more... 1510 words,πŸ•”8 minutes read, May 14, 2022.

Functions of two variables.

 Functions of two variables. Definition, examples, domain, graphs, contour plots. Read more... 1795 words,πŸ•”9 minutes read, May 14, 2022.

Least Squares Interpolation

 Least Squares Interpolation. Fitting a linear and a quadratic model. Moore's Law. Read more... 1812 words,πŸ•”9 minutes read, May 14, 2022.

Partial derivatives

 Definition. Formal Definition. Geometric interpretation. Solved examples. Local extrema and critical points in multivariable functions. Read more... 2114 words,πŸ•”10 minutes read, May 14, 2022.

Tangent Planes And Linear Approximations

 Tangent Planes And Linear Approximations Read more... 1969 words,πŸ•”10 minutes read, May 14, 2022.

Velocity and Acceleration.

 Definition and Properties of Vectors in Motion. Kepler's Laws and Planetary Motion. Newton's explanation using Vector Calculus. Solved exercises. Read more... 3595 words,πŸ•”17 minutes read, May 14, 2022.

Second derivative test

 Quadratic Functions and Critical Points. Steps to Perform the Second Derivative Test. Non-rigorous proof. Solved examples. Read more... 3505 words,πŸ•”17 minutes read, May 14, 2022.

Total differential

 Total differential. Using Different Notation. The Chain Rule for multivariable functions. Proof of the Chain Rule. Solved exercises. Read more... 1378 words,πŸ•”7 minutes read, May 14, 2022.

Directional derivatives

 Step to Compute the Directional derivative. Formal and alternative definition. Solved exercises. Read more... 3364 words,πŸ•”16 minutes read, May 14, 2022.

Gradient vector

 The Gradient Vector is Perpendicular to Level Surfaces. Solved exercises. Read more... 3484 words,πŸ•”17 minutes read, May 14, 2022.

Lagrange Multipliers

 Lagrange Multipliers. Solved exercises. Read more... 3917 words,πŸ•”19 minutes read, May 14, 2022.

Double integrals

 Double integrals. Formal and informal definition, examples, and properties. Calculating a double integral. Fubini's Theorem. Read more... 2765 words,πŸ•”13 minutes read, May 14, 2022.

Double integrals II

 Double integrals II. Solved examples. Exchanging order of integration. Read more... 2292 words,πŸ•”11 minutes read, May 14, 2022.

Implicit partial differentiation

 Calculating βˆ‚f/βˆ‚x, βˆ‚f/βˆ‚y for a given implicit function Read more... 1624 words,πŸ•”8 minutes read, May 14, 2022.

Partial Derivative with Constrained Variables

 Non-independent variables. Solving a Constraint. Avoiding confusion. Calculate the area of a triangle. Read more... 3756 words,πŸ•”18 minutes read, May 14, 2022.

Partial differential equations

 Classification of differential equations. Ordinary Differential Equations. The heat and the harmonic oscillator equations. First-order differential equations. Read more... 3003 words,πŸ•”15 minutes read, May 14, 2022.

Applications of Double Integrals

 Area Region. Volume under Surface. Average Value Function. Total mass of a flat object (lamina) over a region. Read more... 4029 words,πŸ•”19 minutes read, May 14, 2022.

Applications of Double Integrals II

 Weighted Average of a Function with a Density Function. Center of Mass. Moment of Inertia. Kinetic Energy of a Rotating Object. Read more... 4441 words,πŸ•”21 minutes read, May 14, 2022.

Double integrals in Polar Coordinates

 Double integrals in Polar Coordinates Read more... 3372 words,πŸ•”16 minutes read, May 14, 2022.

Change of variables in double integrals

 Understanding the Jacobian in Change of Variables. Applying the Change of Variables Formula. Transformations with polar coordinates. Read more... 4074 words,πŸ•”20 minutes read, May 14, 2022.

Vector fields

 Vector fields Read more... 2960 words,πŸ•”14 minutes read, May 14, 2022.

Conservative vector fields

 Open, connected, and simply connected regions. The Fundamental theorem of Calculus for Line Integral. Equivalent Properties of Conservative Vector Fields. Read more... 2931 words,πŸ•”14 minutes read, May 17, 2022.

Conservative vector fields II

 Path Independence and Conservative Vector Fields. Criterion for a Conservative Vector Field. Curl and Torque Read more... 2620 words,πŸ•”13 minutes read, May 17, 2022.

Find potential functions for conservative fields

 Finding a potential function for conservative vector fields. Solved exercises. Read more... 3297 words,πŸ•”16 minutes read, May 17, 2022.

Green's theorem

 Green's theorem. Fully Explained. Step by Step examples. Read more... 3678 words,πŸ•”18 minutes read, May 17, 2022.

Surface Integrals of Vector Fields. Flux.

 Surface Integrals of Vector Fields. Flux Form of Green's Theorem. Explanation of divergence. Limitations of Green's Theorem. Read more... 3960 words,πŸ•”19 minutes read, May 18, 2022.

Surface Integrals of Vector Fields. Flux. II

 Example with a Non-Simply Connected Region. Definition and Importance of Simply Connected Regions in Green's Theorem. Why Simply Connected Regions Matter. Criteria for Conservative Fields. Solved mixed exercises. Read more... 3257 words,πŸ•”16 minutes read, May 18, 2022.

Triple Integrals

 Triple Integrals. Solved Exercises. Cylindrical Coordinates. Read more... 3410 words,πŸ•”17 minutes read, May 17, 2022.

Triple Integrals 2. Applications.

 Triple Integrals. Solved Exercises. Applications. Read more... 4916 words,πŸ•”24 minutes read, May 17, 2022.

Triple Integrals 3. Spherical coordinates

 Spherical coordinates. Solved Exercises. Applications. Calculation of Gravitational Force Exerted by an object. Read more... 4701 words,πŸ•”23 minutes read, May 17, 2022.

Vector fields & flux 3D

 Vector fields in 3D. Flux in 3D. Solved Exercises. Read more... 4767 words,πŸ•”23 minutes read, May 17, 2022.

Vector fields & Flux in 3D II

 Vector fields & flux in 3D. Solved Exercises. Explanation of Surface Area Elements Using Normal Vectors. Read more... 5023 words,πŸ•”24 minutes read, May 17, 2022.

The Divergence Theorem

 The Divergence Theorem. Examples, Physical Interpretation, and Proof. Step by Step Solved Exercises. Read more... 4926 words,πŸ•”24 minutes read, May 17, 2022.

The Divergence Theorem II

 The Divergence Theorem. Solved Exercises. The diffusion equation Read more... 4220 words,πŸ•”20 minutes read, May 17, 2022.

Curl in 3D

 Curl in 3d, definition and solved exercises. Conditions for a Vector Field to be Conservative. Read more... 2752 words,πŸ•”13 minutes read, May 17, 2022.

Line integrals in space

 Line integrals in space. Solved Exercises. Test for conservative fields. Find potential functions. Read more... 4028 words,πŸ•”19 minutes read, May 17, 2022.

Stoke's Theorem

 Formal Statement, Proof of Stoke's Theorem. Orientation and the Right-Hand Rule. Intuitive explanation. Solved Exercises. Comparing Stokes’ Theorem with Green’s Theorem. Read more... 3433 words,πŸ•”17 minutes read, May 17, 2022.

Stoke's Theorem II

 Stoke's Theorem II. Solved Exercises. Read more... 2699 words,πŸ•”13 minutes read, May 17, 2022.

Stoke and Surface Independence. Curl.

 Solved examples. Stoke Theorem and Surface Independence. Curl and Topological considerations. Physical interpretation of Curl. Conservative Fields and Rotation. Irrotational and Conservative Fields. Read more... 4637 words,πŸ•”22 minutes read, May 17, 2022.

Faraday's law. Maxwell's equations

 Faraday's law. Maxwell's equations. Read more... 1630 words,πŸ•”8 minutes read, May 17, 2022.

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