# Expressing general rotation in terms of tensors

• CAF123
In summary, the general rotation through angle ##a## about the axis ##\underline{n}##, where ##\underline{n}^2 = 1##, is given by $$R(a,\underline{n}) = \exp(-ia\underline{n} \cdot \underline{T}),$$ where ##(T_k)_{ij} = -i\epsilon_{ijk}##. By expanding the exponential as a power series in ##a## and explicitly summing the resulting series, we can show that $$R_{ij}(a,\underline{n}) = \delta_{ij}\cos a + n_i n_j (1-\cos a) - \epsilon_{ijk} n_k \sin a$$ using ##(\underline
CAF123
Gold Member

## Homework Statement

A general rotation through angle ##a## about the axis ##\underline{n}##, where ##\underline{n}^2 = 1## is given by $$R(a,\underline{n}) = \exp(-ia\underline{n} \cdot \underline{T}),$$ where ##(T_k)_{ij} = -i\epsilon_{ijk}##. By expanding the exponential as a power series in ##a##, and explicitly summing the resulting series, show that $$R_{ij}(a,\underline{n}) = \delta_{ij}\cos a + n_i n_j (1-\cos a) - \epsilon_{ijk} n_k \sin a$$ using ##(\underline{n} \cdot \underline{T})_{ij}^2 = \delta_{ij} - n_i n_j## and ##(\underline{n} \cdot \underline{T})_{ij}^3 = (\underline{n} \cdot \underline{T})_{ij}##, (which I derived earlier)

## Homework Equations

##\exp(ix) = \cos x + i\sin x## and cos and sin power series.

## The Attempt at a Solution

$$R(a,\underline{n})_{ij} = [\cos(a (\underline{n} \cdot \underline{T})) - i\sin(a (\underline{n} \cdot \underline{T}))]_{ij}$$ Reexpressing in terms of the Taylor series for sin and cos: $$\left[\sum_{k=0}^{\infty} (-1)^k \frac{(a (\underline{n} \cdot \underline{T}))^{2k}}{(2k)!} - i \sum_{k=0}^{\infty} (-1)^k \frac{(a (\underline{n} \cdot \underline{T}))^{2k+1}}{(2k+1)!}\right]_{ij}$$ Reorganising the even and odd terms gives $$\left[\sum_{k=0}^{\infty} (-1)^k \frac{(\delta_{ij} - n_i n_j)^k}{(2k)!} - i \sum_{k=0}^{\infty} (-1)^k\frac{[(\delta_{ij} - n_i n_j)^{2k} (-i\epsilon_{ijk}n_k)]}{(2k+1)!}\right]$$

I am not really sure how to progress. Thanks for any help.

Hi CAF123!

(btw, you missed out all the aks
CAF123 said:
[… using ##(\underline{n} \cdot \underline{T})_{ij}^2 = \delta_{ij} - n_i n_j## and ##(\underline{n} \cdot \underline{T})_{ij}^3 = (\underline{n} \cdot \underline{T})_{ij}##, (which I derived earlier)

sooo you should have no powers of ##(\delta_{ij} - n_i n_j)## or of ##(\underline{n} \cdot \underline{T})_{ij}## above 1 !

Hi tiny-tim,

I was able to progress and obtained for ##k \geq 1##, ##(\underline{n} \cdot \underline{T})_{ij}^{2k} = \delta_{ij} - n_in_j## and for ##k \geq 0##, ##(\underline{n} \cdot \underline{T})_{ij}^{2k+1} = (\underline{n} \cdot \underline{T})_{ij}##.

For the case ##k=0## in ##(\underline{n} \cdot \underline{T})_{ij}^{2k}##, I was wondering if this should be ##\delta_{ij}##. If so, then I have at the end $$\delta_{ij} + (\delta_{ij} - n_i n_j) (\cos a - 1) - \epsilon_{ijl} n_l \sin a = \text{result}$$ My reasoning being that for all other powers of ##k=0## the result is a matrix, so for ##k=0##, the result should also be a matrix, (= the identity, which is naturally ##\delta_{ij}##.) Thanks.

Hi CAF123!
CAF123 said:
… My reasoning being that for all other powers of ##k=0## the result is a matrix, so for ##k=0##, the result should also be a matrix, (= the identity, which is naturally ##\delta_{ij}##.)

Yes, A0 for any matrix is the identity, I,

so (A0)ij = δij

1 person

## 1. What is a tensor and how is it used in expressing general rotation?

A tensor is a mathematical object that describes the relationship between different coordinate systems. In the context of expressing general rotation, tensors are used to represent the transformation of vectors and other quantities between different coordinate systems, such as Cartesian and spherical coordinates.

## 2. How does expressing general rotation in terms of tensors differ from other methods?

Expressing general rotation in terms of tensors allows for a more elegant and compact representation of rotations compared to other methods such as matrices or quaternions. This is because tensors can capture the full complexity of rotations, including higher-order rotations and non-commutativity, while still being easily manipulated through standard mathematical operations.

## 3. Can tensors be used to express both rigid and non-rigid rotations?

Yes, tensors can be used to represent both rigid and non-rigid rotations. Rigid rotations are those that preserve the shape and size of an object, while non-rigid rotations involve deformation or stretching of the object. Tensors can capture both types of rotation, making them a versatile tool for expressing general rotation.

## 4. What is the relationship between tensors and the rotation matrix?

The rotation matrix is a specific type of tensor that represents a rotation in three-dimensional space. It is a 3x3 matrix that can be used to transform vectors between different coordinate systems. Tensors, on the other hand, can have any number of dimensions and can represent rotations in higher-dimensional spaces.

## 5. Are there any practical applications of expressing general rotation in terms of tensors?

Yes, expressing general rotation in terms of tensors has many practical applications in fields such as physics, engineering, and computer graphics. It is used in areas such as robotics, computer animation, and fluid dynamics to accurately model and simulate rotations in complex systems.

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