Why is there a difference in notation between vectors? For example, I know that a vector is represented by $a^iv_i$, where $a^i$ is a component, and $v_i$ is a covariant basis vector. A dual vector can be represented by $a_iv^i$, where $a_i$ is a component and $v^i$ is a contravariant basis vector; but what is represented by ${a_i}{v_i}$? Is that just a misuse of notation that early calculus classes use?
 A: This notation is known as Einstein's summation notation. I must say, I would be surprised if you see this in early calculus class. It is only after we have seen many times in tensor calculus that the summation dummy index $i$ only appears exactly twice, once in upper index, once in lower index, that we decide to drop the summation sign $\sum_i$. Forcing early calculus students to accept the notation does not simplify anything.
To the topic, when you try to take inner product of two vectors $x^ie_i$ and $y^ie_i$, some people may mistakenly write the inner product as $x^iy^i$. But Einstein summation notation makes it explicit that summation only occurs when the index $i$ appears once in the superscript, and once in the subscript. This one $x^iy^i$ is not ok.
The correct way to write the inner product of two vectors in Einstein summation notation is rather $x^i\delta_{ij}y^j$, where $\delta_{ij}$ is the Kronecker delta, representing the canonical inner product.
A: It isn't really a misuse of notation.  When working with linear(!) $\mathbb{R}^n$ there is no real need to distinguish between contravariant and covariant (since they transform in the same way with respect to an orthogonal linear change of coordinates).  The need to distinguish upstairs and downstairs indices came with general transformation of coordinates and whether you contract with the denominator or the numerator of the Jacobian $\dfrac{\partial (x')}{\partial (x)}$, which isn't relevant or visible in calculus classes.
