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Comment by eball on Indefinite Integral Constant Multiple Rule

Agreed, duplicate, sorry! Thanks for the link

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Comment by eball on Notation: Functions of Derivatives of Variables

I think I meant "function that is a function of its variables derivative". Overall, I think what I'm asking is if it makes sense to write $f(x)=\dot{x}$. From your answer, it sounds like no, but I'm...

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Comment by eball on Notation: Functions of Derivatives of Variables

That's a good point. Here's an example. If I'm trying to define the force $f$ due to friction, I would say something like $f(\dot{x})=-k\dot{x}$. But I could also write $f(x)=-k\frac{dx}{dt}=-k\dot{x}$...

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Comment by eball on Laplace's Equation Coordinate Change

Can you explain why $\ref{1}$ is incorrect? It seems to be the same thing you wrote except you showed how $q_1$ and $q_2$ are explicit functions of $x$ and $y$. Also, were you in agreement that...

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Comment by eball on Partial Differential Equation Change of Variables

Thanks for the reply! I was looking for something more detailed and pedantic about notation.

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Comment by eball on Linearization of the Derivative Block (MATLAB)

@Royi, thank you for the link. That helps a little but it still doesn't get at the heart of my question. It doesn't explain how the block is used to "approximate the linear behavior" and what that...

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Fundamental Theorem of Calculus from Leibniz Rule Applied to Velocity

I am trying to simplify Leibniz Rule to the (first) Fundamental Theorem of Calculus (FTC) but believe I am doing so incorrectly. Leibniz rule can be written as:$$\frac{d}{dt} \int_{f(t)}^{g(t)}...

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The Dual of a Vector Space (Dot Product) Relating to Spatial Vector Algebra

From Rigid Body Dynamic Algorithms by Roy Featherstone:The Dual of a Vector Space: Let $V$ be a vector space. Its dual, denoted$V^∗$, is a vector space having the same dimension as $V$ , and having the...

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Answer by eball for Change of Variables Under Differentiation: A Derivation...

I think I answered my own question while writing it. Using the relationship $\mu=cos(\theta)$ we can write$$d\theta=\tag{5}\frac{d\mu}{sin(\theta)}$$which means (3) can be written...

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Change of Variables Under Differentiation: A Derivation from Karamcheti's...

I'm reading "Principles of Ideal Fluid Aerodynamics" by Karamcheti. On page 285 he starts to construct the solution of$$\tag{1}\frac{\partial}{\partial r}\left(...

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Partial Differential Equation Change of Variables

Problem StatementI have a partial differential equation given by$$\frac{\partial f}{\partial x}=0\tag{1}\label{1}$$where $f=f(x,y)$. If we introduce a new set of coordinates $q_1$ and $q_2$ such...

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Answer by eball for Partial Differential Equation Change of Variables

I think I am trying to find notation that made sense and I think I figured out how to do it using this post.Start by defining the functions we are using$$f: \Bbb{R}^2 \to \Bbb{R}$$$$g: \Bbb{R}^2 \to...

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Comment by eball on Notation for partial derivative of functions of functions

Pertaining to my first point, you can't hold $x(t)$ constant because $x$ is a function of $t$ (i.e. $x=x(t)$).

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Comment by eball on Notation for partial derivative of functions of functions

Let us continue this discussion in chat.

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Comment by eball on Notation for partial derivative of functions of functions

peek-a-boo, thank you for the very thorough response. I truly appreciate you being pedantic with notation. I think you were spot on in where my misconceptions lay.

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