3.3: 4 SolutionsĮxtra (optional) review/catch up session. Wedge products, differential forms and their integrals SolutionsĮxistence of solutions to ODE, Review of course so far. Sec.2.6:0įinish proof of inverse function theorem. Properties of derivative, partial derivative. Homework (due Monday 18-11): Problems sec 2.1: 9 and sec 2.2: 2 Solutions Introduction, Linear algebra, derivative. The Tuesday problem session is for students whose surname starts with A-M and the Wednesday problem session is for students whose name starts with L-Z. There are lectures on Mondays 5115.0317 and Fridays 5118.-156 and problem sessions on Tuesdays 5114.0004 and Wednesdays 5115.0317. All the results mentioned in this paragraph are special cases of this powerful theorem. Taking the $dx$ part of our integrands seriously clarifies all formulas and shows the way to a general fundamental theorem of calculus that works in any dimension, known as the (generalized) Stokes theorem. In the example above this means passing from f(x) to the differential form f(x)dx. The key to understanding this question is to pass from functions to differential forms. How can one make sense of these and are there any more such theorems perhaps in higher dimensions?
What if our function depends on two or more variables? In two and three dimensions, vector calculus gives some partial answers involving div, grad, curl and the theorems of Gauss, Green and Stokes. To introduce the second question, recall what the fundamental theorem of calculus says that the integral of the derivative is the same as the 'integral' of the function on the boundary. For ordinary differential equations we will prove a similar result on the existence and uniqueness of solutions. This is known as the implicit function theorem. One of the main results is that the linearization of the equation predicts the number of solutions and approximates them well locally.
In other words, everything will be differentiable. The key assumption is that everything we do can locally be approximated by linear functions. The approach will be mostly theoretical, schetching a framework in which one can predict how many solutions there will be without necessarily solving the equation. The equations we will address are systems of non-linear equations in finitely many variables and also ordinary differential equations. The material is focused on answering two basic questions:ġ) How to solve an equation? How many solutions can one expect?Ģ) Is there a higher dimensional analogue the fundamental theorem of calculus? Can one find a primitive? The goal of this course is to explore the notions of differentiation and integration in arbitrarily many variables. Please send all your email to Course description Lecturer Roland van der Veen, assistants: Ruben Ijpma and Oscar Koster Multivariable Analysis 2019 Multivariable analysis
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