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So, i was studying my Computer Vision lecture notes and i came across this formula which says Say, i have a function $f(x,y,t)$, $x,y$ and $t$ are the varying factors After $t+ \nabla t$, i have $f(x+\nabla x,y+\nabla y,t+\nabla t)$. Now, taylor series expansion of $f(x,y,t) = f(x,y,t) + \frac{\nabla f}{\nabla x}dx + \frac{\nabla f}{\nabla y}dy + \frac{\nabla f}{\nabla t}dt$

I didn't understand how he arrived at the expansion. Please can anyone explain me or give a link which can help me to understand the above expansion.

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try to find out taylor expansion $f(x)$
$$f(x)=f(x_0)+(x-x_0)f'(x_0)/1 +(x-x_0)^2f''(x_0)/2!+...$$ for $$f(x,y)=f(x_0,y_0)+(x-x_0)\frac{\partial f(x_0,y_0)}{\partial x}+(y-y_0)\frac{\partial f(x_0,y_0)}{\partial y}+1/2![(x-x_0)^2 \frac {\partial ^2 f(x_0,y_0)}{\partial x^2}+(x-x_0)(y-y_0)\frac {\partial ^2 f(x_0,y_0)}{\partial xy}+(y-y_0)^2\frac {\partial^2 f(x_0,y_0)}{\partial y^2}]$$ eventually you can calculate for $f(x,y,t)$

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  • $\begingroup$ :- Thanks a lot. Makes a lot of sense. $\endgroup$ – Raj Mar 12 '16 at 17:38

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