In the mathematical field of geometric measure theory, the coarea formula expresses the integral of a function over an open set in Euclidean space in terms of integrals over the level sets of another function. A special case is Fubini's theorem, which says under suitable hypotheses that the integral of a function over the region enclosed by a rectangular box can be written as the iterated integral over the level sets of the coordinate functions. Another special case is integration in spherical coordinates, in which the integral of a function on Rn is related to the integral of the function over spherical shells: level sets of the radial function. The formula plays a decisive role in the modern study of isoperimetric problems. For smooth functions the formula is a result in multivariate calculus which follows from a change of variables. More general forms of the formula for Lipschitz functions were first established by Herbert Federer, and for BV functions by . A precise statement of the formula is as follows. Suppose that Ω is an open set in and u is a real-valued Lipschitz function on Ω. Then, for an L1 function g, where Hn−1 is the (n − 1)-dimensional Hausdorff measure. In particular, by taking g to be one, this implies and conversely the latter equality implies the former by standard techniques in Lebesgue integration. More generally, the coarea formula can be applied to Lipschitz functions u defined in taking on values in where k ≤ n. In this case, the following identity holds where Jku is the k-dimensional Jacobian of u whose determinant is given by (Wikipedia).
Polar Coordinates Formula In this video, I present a cool version of polar coordinates that is not taught in class. It is not only more elegant, but it also makes calculations with polar coordinates much more elegant. It can even be generalized to what is called the coarea formula. Enjoy!
From playlist Real Analysis
Surface area of sphere in n dimensions
In this sequel to the video "Volume of a ball in n dimensions", I calculate the surface area of a sphere in R^n, using a clever trick with the Gaussian function exp(-1/2 |x|^2). Along the way, we discover the coarea formula, which is the analog of polar coordinates, but in n dimensions. Fi
From playlist Cool proofs
In this sequel to the Laplace-video, I solve Poisson’s equation by showing that Phi convolved with f solves the PDE (where Phi is the fundamental solution of Laplace's equation). Along the way we discover the coarea/onion formula, as well as a n-dimensional version of integration by parts.
From playlist Partial Differential Equations
C07 Homogeneous linear differential equations with constant coefficients
An explanation of the method that will be used to solve for higher-order, linear, homogeneous ODE's with constant coefficients. Using the auxiliary equation and its roots.
From playlist Differential Equations
I continue the look at higher-order, linear, ordinary differential equations. This time, though, they have variable coefficients and of a very special kind.
From playlist Differential Equations
The Definition of a Linear Equation in Two Variables
This video defines a linear equation in to variables and provides examples of the different forms of linear equations. http://mathispower4u.com
From playlist The Coordinate Plane, Plotting Points, and Solutions to Linear Equations in Two Variables
Derive the Auxiliary Equation for a Cauchy-Euler Equation
This video derives the auxiliarly or characteristic equation used to solve a second order Cauchy-Euler differential equation. Site: http://mathispoweru4.com
From playlist Second Order Homogeneous Cauchy-Euler Differential Equations
Ex: Given a Solution to a Differential Equation, Find the Particular Solution
This video explains how to determine a particular solution to a differential equation given the general solution and initial conditions. http://mathispower4u.com
From playlist Introduction to Differential Equations
What is the definition of scientific notation
👉 Learn about scientific notations. Scientific notation is a convenient way of writing very large or very small numbers. A number written in scientific notation is of the form a * 10^n where a is the first non-zero number between 1 and 10, (1 included) and n is the number of digits up to t
From playlist Scientific Notation | Learn About
How to determine if an equation is a linear relation
👉 Learn how to determine if an equation is a linear equation. A linear equation is an equation whose highest exponent on its variable(s) is 1. The variables do not have negative or fractional, or exponents other than one. Variables must not be in the denominator of any rational term and c
From playlist Write Linear Equations
Euler's formulas, Rodrigues' formula
In this video I proof various generalizations of Euler's formula, including Rodrigues' formula and explain their 3 dimensional readings. Here's the text used in this video: https://gist.github.com/Nikolaj-K/eaaa80861d902a0bbdd7827036c48af5
From playlist Algebra
In this video we discuss First-Order Linear Differential Equations. We start by defining what we mean by a Linear Differential Equation and give the formula for the integrating factor. Then we do two examples where we carefully go over the steps. 0:00 Introduction to Linear Differential E
From playlist Linear Differential Equations
Excel Basics 12: Complete Formula Lesson of Formula Types & Formula Elements 12 Examples
Highline College Busn 216 Class: Computer Applications taught by Michael Girvin. Download Excel Start & Finished File: https://people.highline.edu/mgirvin/YouTubeExcelIsFun/EB12-FormulaElements.xlsm Download Full Written Notes: https://people.highline.edu/mgirvin/YouTubeExcelIsFun/Office20
From playlist Excel Formulas - Basics and Beyond
Highline Excel 2016 Class 02: Excel Fundamentals: Comprehensive Formula Lesson (15 Examples)
Download Files: https://people.highline.edu/mgirvin/AllClasses/218_2016/218Excel2016.htm In this video learn about: 1. (00:13) Introduction 2. (10:09) Excel’s Golden Rule 3. (04:31) Look at list of Formula Types and Formula Elements 4. (06:29) Number Formula Examples 5. (08:24) Text and Nu
From playlist Excel Advanced Free Course at YouTube. Comprehensive Excel 2016: Calculations & Data Analysis (27 Videos)
Excel Statistical Analysis 03: Comprehensive Worksheet Formula & Model Building Video
Download Excel File: https://excelisfun.net/files/Ch01-ESA.xlsm Learn the fundamentals of Excel worksheet formulas, functions and how to build worksheet models. Topics: (00:00) Introduction (00:19) Topics in video (00:45) Worksheet Formula Elements: what can go into a formula (01:57) Excel
From playlist Excel Statistical Analysis for Business Class Playlist of Videos from excelisfun
Highline Excel 2013 Class Video 02: Comprehensive Excel Formula Types & Elements Video 12 Examples
Download workbook: http://people.highline.edu/mgirvin/AllClasses/214_2013/214/Busn214_2013.htm This is the Highline Community College Class, Spreadsheet Construction taught by Michael Girvin: Basic To Advanced Excel. Topics in this video: Overview: (00:39 min) 1. Hours worked formula. Type
From playlist Excel Formulas - Basics and Beyond
Ctrl + Shift + Enter: Excel Array Formulas #02: Introduction Array Formulas & Math Array Operation
Download files here: http://people.highline.edu/mgirvin/excelisfun.htm EXCEL ARRAY FORMULAS WORK THE SAME IN ANY VERSION OF EXCEL!!! This video covers: 1. (01:00 min) What is an "Array"? 2. (01:55 min) Arrays in Excel. 3. (02:41 min) What is an Array Formula? (See book for more complete d
From playlist Ctrl+Shift+Enter: Mastering Excel Array Formulas (35+ Videos in Series)
M 365 Excel Worksheet Formulas & Models – Everything You Ever Wanted To Know - 365 MECS 03
Download Excel File: https://excelisfun.net/files/03-M365ExcelClass.xlsx Download PDF File: https://excelisfun.net/files/03-M365ExcelClass.pdf This video is a comprehensive lesson in using Excel Worksheet functions and formulas to build Worksheet Solutions and Models. This full free Micr
From playlist Microsoft 365 Excel Complete Class: free from excelisfun at YouTube - 365 MECS
Excel Formulas & Functions in 10 Minutes. Excel Magic Trick 1629.
Download Excel File: https://excelisfun.net/files/EMT1629.xlsx Learn about the fundamentals of Excel Formulas and Functions in 10 minutes. Excel Workbook for download so you can follow along. Topics: 1. (00:01) Introduction 2. (00:57) Equal Sign Starts all Formulas. 3. (00:57) Create Formu
From playlist Excel Formulas - Basics and Beyond
Differential Equatons: Find the Order and Classify as Linear or Nonlinear
This video explains how to determine the order of a differential equation and how to determine if it is linear or nonlinear. http://mathispower4u.com
From playlist Introduction to Differential Equations