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PHY 101: Vector Integration

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What is Integration? Integration can be thought of as a continuous analogue of sum (\(\sum_{ }^{ }\)). We integrate or simply add infinitesimal small quantities together to form a continuous chain. Integration can be of 3 kinds rather I would call it 3 ways of integration. Linear, Area and Volume way. To show what each of them looks like here is a visual representation: $$\begin{aligned}& Linear \hspace{1mm}integration:\int_{ }^{ }f(x)dx\\ \\ &Area\hspace{1mm}or\hspace{1mm} surface\hspace{1mm} integration: \iint_S f(x,y)dxdy \\ \\ & Volume \hspace{1mm} integration: \iiint_V f(x,y,z)dxdydz \end{aligned}$$ You might have already noticed that the number of integration symbols (\(\int_{ }^{ }\)) increases with the increase in the number of variables. Hence, most books adopt the notation of calling these single, double and triple integrations. We at physics are creatures of simplicity and thus have kept it easy to remember. Let us talk about each in some detail! Single or Linear...
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PHY 101: Vector Integration

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What is Integration? Integration can be thought of as a continuous analogue of sum (\(\sum_{ }^{ }\)). We integrate or simply add infinitesimal small quantities together to form a continuous chain. Integration can be of 3 kinds rather I would call it 3 ways of integration. Linear, Area and Volume way. To show what each of them looks like here is a visual representation: $$\begin{aligned}& Linear \hspace{1mm}integration:\int_{ }^{ }f(x)dx\\ \\ &Area\hspace{1mm}or\hspace{1mm} surface\hspace{1mm} integration: \iint_S f(x,y)dxdy \\ \\ & Volume \hspace{1mm} integration: \iiint_V f(x,y,z)dxdydz \end{aligned}$$ You might have already noticed that the number of integration symbols (\(\int_{ }^{ }\)) increases with the increase in the number of variables. Hence, most books adopt the notation of calling these single, double and triple integrations. We at physics are creatures of simplicity and thus have kept it easy to remember. Let us talk about each in some detail! Single or Linear...
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PHY-101: Vector Differentiation

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Vector Differentiation What is Differentiation? Simply put, Differentiation is the rate of change of a function with respect to a variable. For example, velocity tells us how fast and in which direction an object is moving, which is the rate of change of the object in space with respect to time. Mathematically, $$v=\frac{dx}{dt}$$where x is the space coordinate in 1D and t is time. In a way, mathematicians like to define is, the slope of the line tangent to the graph of \(f(x)\) Fig1: The blue line shows the function f(x) and the red line is the first derivative of the function To understand how to calculate a derivative of a function I'll give you a very basic formula widely known as the First principle $$f '(x)=\lim_{\Delta x\to0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$$where \(f '(x)\) is the first derivate, \(f(x)\) is the function and \(\Delta x\) is the change we are calculating the function over. You might encounter a few notations while dealing with derivatives of a fun...
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