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fresh_42 submitted a new PF Insights post
The Pantheon of Derivatives - Part II
Continue reading the Original PF Insights Post.
The Pantheon of Derivatives - Part II
Continue reading the Original PF Insights Post.
... which (is not claimed and) emphasizes my point to watch out the definitions made in the relating contexts.zwierz said:the space ##C^\infty_{\mathbb{R}}([0,1])## equipped with the uniform norm is not complete. That is not the point but above it is written that all the spaces are Banach
True. I removed ##\mathbb{R}^n##.The definition of a vector field on a manifold is very bad. It looks like one implicitly assumes that the manifold is embedded into ##\mathbb{R}^m##.
It is a line bundle, and the set ##\{(x,\nabla_xf)\,\vert \,x \in M\}## is a vector field.the gradient of a function is not a vector field, it is an 1-form at least until the manifold is not equipped with a metric tensor
##\vec{a} \times \vec{b}## is a vector in ##\mathbb{R}^3##.the operation rotor as well as cross product give pseudo vectors
So can I. But what for? Your goal is apparently one which I'm not interested in.zwierz said:In principle I can dig textbooks up and show you corresponding paragraphs but what for?
It would be great if you ask a specialist in differential geometry whom you trust in , to read your text carefully.
I'll have it in the fourth part (where I also included an example from Wiki): The material derivative is a special case of a derivative in order to describe the flow of fluids or gases. It is more of a special tool for these currents rather than a special concept of a differentiation process.Stephen Tashi said:Is the "material derivative" of fluid mechanics a special case of a Lie derivative? Or is it yet another kind of derivative?
Good question.Stephen Tashi said:Is the "material derivative" of fluid mechanics a special case of a Lie derivative? Or is it yet another kind of derivative?
The Pantheon of Derivatives is a term used to describe the collection of all the different types of financial derivatives that exist in the market. These derivatives are financial instruments that derive their value from an underlying asset, such as stocks, bonds, commodities, or currencies.
Part II - Comments is a section within the Pantheon of Derivatives that allows for discussion and analysis of the various types of derivatives. It provides a platform for experts and individuals to share their thoughts, opinions, and insights on the different derivatives and their impact on the market.
Derivatives are used in the financial market for a variety of purposes, such as hedging against risk, speculating on future market movements, and managing portfolio risk. They can also be used to create leverage and increase potential returns on investments.
Some common types of derivatives include options, futures, forwards, swaps, and credit derivatives. Options give the holder the right, but not the obligation, to buy or sell an underlying asset at a predetermined price. Futures are contracts to buy or sell an asset at a specified price on a future date. Forwards are similar to futures, but they are not traded on an exchange. Swaps involve the exchange of cash flows between two parties based on the performance of an underlying asset. Credit derivatives allow investors to hedge against credit risk.
Derivatives can involve significant risks, such as market risk, credit risk, liquidity risk, and counterparty risk. Market risk refers to the potential for losses due to changes in the market value of the underlying asset. Credit risk is the risk of default by the counterparty in a derivative transaction. Liquidity risk is the risk of not being able to sell a derivative quickly at a fair price. Counterparty risk is the risk of the other party in a derivative transaction not fulfilling their obligations.