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Loïc Guégan 2024-08-28 20:55:15 +02:00
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@ -5,17 +5,27 @@ Differentiation Notations
Leibniz's Notation Leibniz's Notation
------------------- -------------------
The derivative of a function :math:`f` at :math:`x` is given The derivative of a function :math:`f` at :math:`x` is given by :math:`\lim\limits_{h\to0} \frac{f(x+h)-f(x)}{h}`.
by :math:`\lim\limits_{h\to0} \frac{f(x+h)-f(x)}{h}`.
The Leibniz's notation expresses the derivative of :math:`f` as :math:`\frac{dy}{dx}` with :math:`y=f(x)`. The Leibniz's notation expresses the derivative of :math:`f` as :math:`\frac{dy}{dx}` with :math:`y=f(x)`.
More details `here <https://www.me.psu.edu/cimbala/me420/Homework/dydx_quotient_article.html>`__. More details `here <https://www.me.psu.edu/cimbala/me420/Homework/dydx_quotient_article.html>`__.
.. note:: .. note::
The following :math:`\frac{dx}{dt}` means :math:`x` is a function of :math:`t` such as :math:`x=f(t)`. The following :math:`\frac{dx}{dt}` means :math:`x` is a function of :math:`t` such as :math:`x=f(t)`.
See explanations `here <https://www.khanacademy.org/math/ap-calculus-ab/ab-diff-contextual-applications-new/ab-4-4/v/differentiating-related-functions-intro>`__. See explanations `here <https://www.khanacademy.org/math/ap-calculus-ab/ab-diff-contextual-applications-new/ab-4-4/v/differentiating-related-functions-intro>`__.
:math:`\frac{d}{dx}` is an operator not a quotient!
Although it behaves like a quotient. In fact it is a limit:
.. math::
\lim\limits_{h\to0} \frac{f(x+h)-f(x)}{h} \ne \frac{\lim\limits_{h\to0} \left(f(x+h)-f(x)\right)}{\lim\limits_{h\to0} h}
See, we cannot express this limit-of-a-quotient as a-quotient-of-the-limits, then the derivative is not a quotient.
More details `here <https://www.me.psu.edu/cimbala/me420/Homework/dydx_quotient_article.html>`__.
Can we perform operation without relying on this assumption?
The answer is yes! Using the chain rule.
Lagrange's Notation Lagrange's Notation
------------------- -------------------
Also cal prime notation Also cal prime notation

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Gauss's Law
============

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The Maxwell's Equations
========================================
.. toctree::
:maxdepth: 2
:caption: First equation
gauss.rst

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@ -33,6 +33,7 @@ Welcome to ScienceNotes's documentation!
:numbered: :numbered:
:caption: Electronics :caption: Electronics
electronics/maxwell/index.rst
electronics/metrics.rst electronics/metrics.rst
electronics/components.rst electronics/components.rst
electronics/kirchhoff.rst electronics/kirchhoff.rst