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Try to make things strictly correct, even if reader doesn't understand just why. It's really more straightforward. In progress.
'''Tensor index notation''' is a method of notation which is of use when dealing with [[tensor]]s.<ref>[http://www.brown.edu/Departments/Engineering/Courses/En221/Notes/Index_notation/Index_notation.htm Tensor index notation] from brown.edu</ref> Particular examples of tensors include [[vector]]s and [[matrix|matrices]], and index notation can greatly simplify algebraic manipulations involving these types of mathematical object.
'''Tensor index notation''' is a method of notation which is of use when dealing with [[tensor]]s.<ref>[http://www.brown.edu/Departments/Engineering/Courses/En221/Notes/Index_notation/Index_notation.htm Tensor index notation] from brown.edu</ref> Particular examples of tensors include [[vector]]s and [[matrix|matrices]], and index notation can greatly simplify algebraic manipulations involving these types of mathematical object.
The components of a vector (with respect to some co-ordinate system) might be written <math>\boldsymbol{x}=(x_1,x_2,x_3)</math>. More concisely, we could write <math>x_i</math> for the components of the vector, where <math>i=1,2,3</math>. To motivate this notation, we will consider the equation <math>Ax=b</math> for some matrix <math>A</math> and vectors <math>x,b</math>. We will use the convention that if <math>A</math> is a matrix, then <math>(A)_{ij}=a_{ij}</math> is the element of that matrix in the ith row and jth column. One way of thinking of vector equations is as a shorthand for a set of simultaneous equations - each component of the vectors gives an equation. Explicitly, consider the set of three equations for the three unknowns <math>x_1, x_2, x_3</math>:
The components of a vector (with respect to some co-ordinate system) might be written <math>\boldsymbol{x}=(x_1,x_2,x_3)</math>.
But before we go any further, we should note that proper tensor index notation is quite fussy about which indices are subscripts and which are superscripts. Elementary treatments don't bother with this, and just use a "subscripts everywhere" philosophy. But proper placement of indices as subscripts and superscripts is important in proper tensor algebra. Even if one doesn't understand the theoretical reasons for this, things are notationally simpler once one gets used to it. The fundamental principle, for now, is that:
:'''Indices of vectors are written as superscripts.'''
Just get used to that.
So, the components of the above vector could be written <math>\boldsymbol{x}=(x^1,x^2,x^3)</math>.
More concisely, we could write <math>x^i</math> for the components of the vector, where <math>i=1,2,3</math>. To motivate this notation, we will consider the equation <math>Ax=b</math> for some matrix <math>A</math> and vectors <math>x</math> and <math>b</math>. We will use the convention that if <math>A</math> is a matrix, then <math>(A)_{ij}=a_{ij}</math> is the element of that matrix in the ith row and jth column. One way of thinking of vector equations is as a shorthand for a set of simultaneous equations - each component of the vectors gives an equation. Explicitly, consider the set of three equations for the three unknowns <math>x^1, x^2, x^3</math>:
:<math>
:<math>
a_{11} x_1+a_{12} x_2+a_{13} x_3 = b_1
a_{11} x^1+a_{12} x^2+a_{13} x^3 = b^1
</math>
</math>
:<math>
:<math>
a_{21}x_1+a_{22}x_2+a_{23}x_3 = b_2
a_{21} x^1+a_{22} x^2+a_{23} x^3 = b^2
</math>
</math>
:<math>
:<math>
a_{31}x_1+a_{32}x_2+a_{33}x_3 = b_3
a_{31} x^1+a_{32} x^2+a_{33} x^3 = b^3
</math>
</math>