# The function f is arbitrary, so we can write d dx a = d d x a in VS .NET Create QR-Code in VS .NET The function f is arbitrary, so we can write d dx a = d d x a

The function f is arbitrary, so we can write d dx a = d d x a
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d In other words, we have expanded the vector d in terms of a set of basis vectors. a x The components of the vector are dd and the basis vectors are given by a . x This is the origin of the argument that the basis vectors ea are given by partial derivatives along the coordinate directions.
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EXAMPLE 3-4 The fact that the basis vectors are given in terms of partial derivatives with respect to the coordinates provides an explanation as to why we can write the transformation matrices as
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SOLUTION 3-4 This can be done by applying the chain rule to a basis vector. We have, in the primed coordinates, ea = = a a x x xb xb = xb = xa xb
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Similarly, going the other way, we have eb = = b b x x xa xa = xa = xb xa
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Now let s explore some new notation that is frequently seen in books and the literature. We can write the dot product using a bracket-type notation , . In the left slot, we place a one form and in the right side we place a vector. And so we write the dot product p v as p v = p, v = pa v a (3.3)
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Here p is a one form. Using this notation, we can write the dot product between the basis one forms and basis vectors as
a a , eb = b
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This can be seen easily writing the bases in terms of partial derivatives: a , eb = dx a , xa a = b = b xb x
This type of notation makes it easy to nd the components of vectors and one forms. We consider the dot product between an arbitrary vector V and a basis one form:
a a , V = a , V b eb = V b a , eb = V b b = V a
Since the components of a vector are just numbers, we are free to pull them outside of the bracket , . We can use the same method to nd the components of a one form:
a , eb = a a , eb = a a , eb = a b = b
Now we see how we can derive the inner product between an arbitrary one form and vector:
a , V = a a , V b eb = a V b a , eb = a V b b = a V a
These operations are linear. In particular, , aV + bW = a , V + b , W a + b , V = a , V + b , V where a, b are scalars, , are one forms, and V, W are vectors.
Tensors as Functions
A tensor is a function that maps vectors and one forms to the real numbers. The components of a tensor are found by passing basis one forms and basis vectors as arguments. For example, we consider a rank 2 tensor T . If we pass two basis one forms as argument, we get T a , b = T ab
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A tensor with raised indices has contravariant components and is therefore expanded in terms of basis vectors; i.e., T = T ab ea eb We have already seen an example of a tensor with lowered indices, the metric tensor gab a b = gab dx a dx b (in a coordinate basis). A tensor is not xed with raised or lowered indices; we recall from the last chapter that we can raise or lower indices using the metric. We can also have tensors with mixed indices. For each raised index we need a basis vector, and for each lowered index we need a basis one form when writing out the tensor as an expansion (the way you would write a vector expanding in a basis). For example S = S ab c ea eb c We get the components in the opposite way; that is, to get the upper index pass a one form, and to get the lower index pass a basis vector: S ab c = S a , b , ec We can pass arbitrary vectors and one forms to a tensor. Remember, the components of vectors and one forms are just numbers. So we can write S ( , , V ) = S a a , b b , V c ec = a b V c S a , b , ec = a b V c S ab c The quantity a b V c S ab c is a number, which is consistent with the notion that a tensor maps vectors and one forms to numbers. Note that the summation convention is used.