Mathematical tools for thermodynamics 1

Name: Mathematical tools for thermodynamics 1
Rating: 4.5 (10 reviews)

For beginners one of the best way to begins thermodynamics, In the form of direct application, with detailed quiz answer

Created byMohamed Elboudour Elidrissi

Last updated 8/2017

English

What you'll learn

Learn polar coordinates
Ability to calculate a partial derivative
Know when a differential is exact or not
Determine a function from its total differential
Know what it is a state equation
Ability to calculate an easy path integral
And in the end it's about training his brain

Course content

4 sections • 32 lectures • 3h 45m total length

Recall about polar coordinates12:33
Explore polar coordinates by linking Cartesian x,y and polar r, theta via x = r cos theta, and y = r sin theta, with r = sqrt(x^2+y^2) and related derivatives.
Polar coordinate
Explanation of quiz 1
First order partial derivative7:17
Explore first order partial derivatives by treating other variables as constants, and learn the standard notation for partial derivatives with practical examples involving r, theta, sine and cosine terms.
Quiz 2
Explanation of quiz 2
Mixed second order partial derivative part 17:53
calculate a mixed second-order partial derivative with respect to r and theta. use the product rule and treat constants to obtain an expression involving cos theta and sin theta.
Mixed second order partial derivative part 28:27
Explore mixed second-order partial derivatives, focusing on derivatives with respect to multiple variables, pattern recognition for product rules, and evaluating trig derivatives like cos and sin with respect to theta.
Quiz 3
Explanation of quiz 3
Recall about derivatives of cos and sin and part 3 of mixed partial derivative11:18
Explore derivatives of cosine and sine using prime and partial notation across single- and multi-variable functions. See mixed partial derivative concepts with cosine squared and sine relations.
Quiz 4
Errata about session 7
Derivative
Explanation of quiz 2
A general result in Thermodynamics11:32
Explore a general result in thermodynamics by examining how mixed second partial derivatives are equal, shown through calculating partial derivatives with respect to theta and using cosine and sine relationships.
Quiz 6
Explanation of quiz 6

Differential form9:21
Learn how differential forms serve as a key mathematical tool in thermodynamics, distinguishing exact differentials from non-exact ones and identifying potential functions for multi-variable systems.
Exact differential form8:44
Evaluate differential forms for exactness by verifying ∂A/∂y = ∂B/∂x in two variables. Clarify notations: use d for derivatives and ∂ for partial derivatives, and distinguish exact from inexact forms.
Quiz 7
Explanation of quiz 7
Exact differential form, part 216:11
Determine when a differential form is exact by checking ∂M/∂y = ∂N/∂x, and relate to total differential dF = M dx + N dy for functions of x and y.
Quiz 8
Explanation of quiz 8
Integral of a function of two variables, part 111:52
Explore when a two-variable differential is exact and find the potential function f(x,y) using the total differential and partials, where f_x = y sin x and f_y = -cos x.
Integral of a function of two variables, part 213:14
Learn to integrate a function of two variables in x while holding y constant, yielding F(x,y) = - y cos x + C(y), where integration constant may depend on y.
Quiz 9
Explanation of quiz 9
Integral of a function of two variables, part 39:58
Examine a two-variable function f(x,y) and verify the constant is independent of y via partial derivatives and exactness, yielding f(x,y) = a − y cos x + c.
Quiz 10

Integral of a function of two variables, other example18:02
Demonstrate that entropy S is a state function of temperature and volume by constructing ds as an exact differential and integrating to S(T,V) with a constant.
Quiz 11
State function11:38
Identify the state function in thermodynamics by treating s as an exact differential; verify conditions with partial derivatives to show dS forms a state function.
Quiz 12

Questions2:07
Compute the line integrals of work along the path A-C-B in the PV plane, relating W1 and W2. Show the PV differentials are exact with a positive constant.
Simple example of path integral, part 19:26
Simple example of path integral, part 29:05
Demonstrate evaluating a simple path integral from a to b, with constant pressure and PV work terms, showing how PDV and DV contributions combine in a particle-like case.
Simple example of path integral, part 19:41
Compute the path integral of work along the piecewise path a→c→b, showing constant-volume segments contribute zero, and express the total work as minus P dV plus PV differences.
Remark10:47
Demonstrate that the line integral w1 is path-independent as an exact differential, and that w2 is inexact, depending on the chosen path from a to b.
Exact differential form, again10:21
Examine exact differential forms and verify path independence by comparing partial derivatives; show w1 is exact with an endpoint-dependent integral, while w2 is inexact and path dependent.
Quiz 13
Exact differential form, again and again5:25
Revisits exact differential forms, applies partial derivatives and exactness tests to identify when a differential is exact, and sets up the next topic.
Exact differential form, again and again and again ( no more after that...)6:40
Explore exact differential forms in thermodynamics, using partial derivatives with respect to P and V to integrate W4, even without full knowledge of W3.
Integrating factor3:52

Requirements

Basic calculus, basic algebra ( like a+a=2a, a*a=a²).
Basic analysis (like derivative of function of one variable).
Basis integral calculus.
High school math level.

Description

This course is reserved for people who have never had a thermodynamics course, or who have never had an advanced course on differential calculus. Anyone can use it just needs a HighSchool Diploma level, this course is presented as a direct application where every detail is explained. Each quiz contains a detailed answer with explanation.

Before any thermodynamics course we need mathematical tools like :

The partial derivative
Mixed second partial derivative
differential form
integral of function of two variables
notion of state function
path integral
integrating factor

In this course we present all this concept for beginner

Our method is detailed the calculation from A to Z .

Content and Overview

About 4 hours of classes using a white board as a real university course for total immersion.

We start with a recall about the polar coordinates followed directly by a quiz with detailed correction, after this we attack the notion of partial derivative of a function of several variables, here, for simplicity we will only see the functions of two variables, but to understand for two variables is to understand automatically for several variables, after that, we will see the mixed derivative.

During our course we will see quite a recall on some rules of derivation already seen in high school.

At the end of section 1 we will see an interesting property in thermodynamics, a general result.

In the second part of this course we shall attack the notion of differential forms, very important concept in thermodynamics, we shall see especially the notion of an exact differential form of a function, since all the functions in thermodynamics their differential is exact.

In the third part we will see the notion of "state function", because all the thermodynamic functions are state functions.

Finally part 4 is a great application of that has been seen previously, it is like a summary, but in this part we will also calculate very simple path integrals for beginners, just to see that there exist such integrals, wich may depend on the path taken.

And at the end we will see an example of an integrating factor, which in fact multiplied by a differential form will allow to integrate, hence its name.

Who this course is for:

Everybody who loves physics
People who study thermodynamics for the first time

Mathematical tools for thermodynamics 1

What you'll learn

Explore related topics

Course content

Partial derivative and mixte second order partial derivative12 lectures • 59min

Exact differential form and Integral of a function of two variables9 lectures • 1hr 9min

State function2 lectures • 30min

Application9 lectures • 1hr 7min

Requirements

Description

Who this course is for: