# Ideal gases and Vector calculus

• bjon-07
In summary, an ideal gas has four properties according to the physics book: nonviscous, steady flow, incompressible, and irrotational. The question arises whether being irrotational is the same as having a curl of zero. The physics book states that irrotational functions have no angular momentum, but the calculus book does not provide a physical definition. It is suggested that incompressibility implies no divergence and irrotational means no curl. However, it is advised to wait for an answer from someone knowledgeable about fluids. Additionally, there is a discrepancy between an ideal gas being compressible while an ideal fluid is defined as incompressible. This can be attributed to a semantic difference, as gases are considered fluids.
bjon-07
In my physics book, the 4 properties of an ideal gase are

1. nonviscous
3. incompressible
4. irrotational

My question is the properties of being irrotional the same as the vector functions that have a Curl=O iff irrotational

My physics book states the irrotional functions have no angular momentum, but my caclulus book does not give a physical defention of an irrotional function, only a mathmatical defention.

So am I right to aqquate the two defentions to gether

bjon07,

This sounds more like the definition of an ideal fluid, than an ideal gas.

But your guess seems right to me. Incompressiblity implies that the flow has no divergence, so irrotational probably means no curl.

But if this is important you should wait until somebody who knows something about fluids has a chance to answer. I don't know much about them; I'm just guessing. ;-)

Oppss, hehe I meant Ideal fluid

1.An ideal gas IS AN IDEAL FLUID.

2.Incompressible,means the density constant,by the law of mass conservation,the divergence of the convective velocity field is zero.

3.Irrotational means that the curl of the convective velocity field is zero.

4.Nonviscous means no friction between neighboring fluid layers.The viscosity tensor is identically zero.The kinetic tension tensor is diagonal and has one independent component,the negative of hydrostatic pressure.

Daniel.

I thought most gase where compressable, pv=nrt...v can be compressed

So what...?

Daniel.

Well, there is an apparent discrepancy here, how can an ideal gas be an ideal fluid when an ideal gas can be compressed, yet an ideal fluid, by definition is incompressible.

To the OP, some spelling suggestions;

gas, not gase
irrotational, not irrotional
calculus, not caclulus
definition, not defention
mathematical, not mathmatical
equate, not aqquate
together, not to gether

Claude.

bjon-07 said:
I thought most gase where compressable, pv=nrt...v can be compressed

It's just a semantic thing. Gases are considered to be fluids (along with liquids). What you're describing at the top is an ideal incompressible fluid. You're right that ideal gases are not incompressible.

Last edited:

## 1. What is an ideal gas?

An ideal gas is a theoretical model of a gas that follows the ideal gas law, which states that the pressure, volume, and temperature of a gas are directly proportional under constant moles and constant energy. In other words, an ideal gas does not have any intermolecular interactions or occupy any volume, and it only exists in theory.

## 2. What is the ideal gas law?

The ideal gas law is a mathematical equation that describes the relationship between the pressure, volume, and temperature of an ideal gas. It is written as PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the absolute temperature.

## 3. What is vector calculus?

Vector calculus is a branch of mathematics that deals with vectors and functions of multiple variables. It involves the study of vector fields, which are functions that assign a vector to each point in space, and the operations of differentiation and integration on these fields. It is widely used in physics and engineering to describe and analyze physical quantities that have both magnitude and direction.

## 4. What is a gradient in vector calculus?

A gradient is a vector that points in the direction of steepest increase of a scalar field. In other words, it represents the direction and magnitude of the maximum rate of change of the scalar field at a given point. It is calculated by taking the partial derivatives of the scalar field with respect to each variable.

## 5. How is vector calculus applied in science?

Vector calculus is used in many scientific fields, such as physics, engineering, and fluid mechanics, to analyze and solve problems involving physical quantities that have both magnitude and direction. It is used to describe the motion of objects, the flow of fluids, and the behavior of electromagnetic fields, among others. It is also used in computer graphics and machine learning to model and manipulate data in multiple dimensions.

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