Comment by eball on Partial Derivatives 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. $x=x(t)$).
View ArticleComment by eball on Partial Derivatives of Functions of Functions
Let us continue this discussion in chat.
View ArticleComment by eball on Partial Derivatives 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.
View ArticleComment by eball on Indefinite Integral Constant Multiple Rule
Agreed, duplicate, sorry! Thanks for the link
View ArticleComment 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...
View ArticleComment 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 $f(x)=-k\frac{dx}{dt}=-k\dot{x}$...
View ArticleComment 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...
View ArticleComment by eball on Partial Differential Equation Change of Variables
Thanks for the reply! I was looking for something more detailed and pedantic about notation.
View ArticleComment 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...
View ArticleFundamental 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:$$\frac{d}{dt} \int_{f(t)}^{g(t)}...
View ArticleThe Dual of a Vector Space (Dot Product) Relating to Spatial Vector Algebra
From Rigid Body Dynamic Algorithms by Roy Featherstone:The Dual of a Vector Space: Let $V$ be a vector space. Its dual, denoted$V^∗$, is a vector space having the same dimension as $V$ , and having the...
View ArticleAnswer by eball for Change of Variables Under Differentiation: A Derivation...
I think I answered my own question while writing it. Using the relationship $\mu=cos(\theta)$ we can write$$d\theta=\tag{5}\frac{d\mu}{sin(\theta)}$$which means (3) can be written...
View ArticleChange of Variables Under Differentiation: A Derivation from Karamcheti's...
I'm reading "Principles of Ideal Fluid Aerodynamics" by Karamcheti. On page 285 he starts to construct the solution of$$\tag{1}\frac{\partial}{\partial r}\left(...
View ArticlePartial 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...
View ArticleAnswer 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$$f: \Bbb{R}^2 \to \Bbb{R}$$$$g: \Bbb{R}^2 \to...
View ArticleAugment State Space Model with Derivative of State
ProblemHow do you augment a state space model with the derivative of a state? I know how to augment a state space model with the integral of a state by doing the following. Given a linear system$$...
View ArticleComment 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. $x=x(t)$).
View ArticleComment by eball on Notation for partial derivative of functions of functions
Let us continue this discussion in chat.
View ArticleComment 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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