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Applied Analysis

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Week Six

If $ g $ is a differentiable function at some point $ P $, then
$ \partial_v g(P) = L\cdot v $ for every vector $ v $ in $ E_n $
such that $ \|v\| = 1 $.

The function $$ g(x,y) := \begin{cases} \frac{xy^2}{x^ 2 + y^2} & \text{ if } x^2 + y^2\neq 0\\ 0 &\text{ if } x = y = 0\end{cases}$$

is continuous, the partial derivatives exist on $ \mathbb{R}^ 2 $ and it is not differentiable at the origin.
Here you can find some exercises.
Please, also do exercises 4, 6 and 12 at page 74 of the book of M. Corral.
The due date is 2014, October 14.
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Week Five

Cartesian forms of lines and planes, intersections of two planes.
Definition of limits for two variables functions.
Here you can find some exercises
(solutions).
The due date is 2014, October 6.

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Week Four

Formulas for the intersection between
two lines in $ \mathbb{R}^3 $ and for the distance between a point and a line and a point and a plane.

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Week Three

The cross product in $ E_2 $. Definition of cross
cross product in $ E_3 $ with the Levi-Civita symbol.
Triple products. Lines in the plane.

Here you can find some exercises on the cross product
(sol.)
The due date is 2014, September 29.
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Week One

The linear structure of $ \mathbb{R}^n $.
Applied vectors of $ \mathbb{R}^n $.
Definition of scalar product. The Cauchy-Schwarz inequality.

Here you can find some exercises on the scalar product
(solutions).
The due date is 2014, September 22.