- #1
I can just barely read the images you posted.granzer said:In
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How is equation 145 giving a direction(ie gradient) and not a slope?.Also here
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how is equation 147 arrived at?
Any help would be much appreciated.
@Mark44 Hello Sir,Mark44 said:I can just barely read the images you posted.
For your first question, in the US, we tend to call ##\frac {dy}{dx}## a slope; in Europe, people tend to call this a gradient. I prefer to reserve the term gradient to functions such as this: If z = f(x, y), then ##\nabla z = (\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y})##
If y = 3x, then y' or (##\frac{dy}{dx}##) = 3. This gives a direction in the sense that from any point on the graph of this line, you can get to another point by going right 1 unit and then up 3 units.
Same idea for your second question.
An inexact differential is a type of differential in mathematics that is not exact. It represents the change in a function that is dependent on multiple variables, but the order in which the variables are changed affects the outcome.
An exact differential is a type of differential in mathematics where the order in which the variables are changed does not affect the outcome. It represents the change in a function that is dependent on multiple variables, but the variables are independent of each other.
Inexact differentials are important in fields such as thermodynamics, where the change in a system depends on multiple variables, and the order in which the variables are changed can affect the outcome. Understanding inexact differentials allows scientists to accurately model and predict the behavior of complex systems.
Inexact differentials are often represented using partial derivatives, denoted by ∂, to indicate that the variables are independent of each other. For example, a partial derivative of a function f(x,y) with respect to x would be denoted as ∂f/∂x.
One example of an inexact differential is work done on a gas in thermodynamic systems. The work done depends on both the pressure and volume of the gas, but the order in which these variables are changed can affect the outcome. Therefore, the work done is an inexact differential.