MTH3008 Lecture 5

Having covered the combinations of the Gradient, Divergence, and Curl in previous lectures, we now look at how these operators apply to products of functions, and then begin exploring coordinate systems and transformations.

Operations on Products of Functions

Using Suffix Notation, we can compute operations on products of functions, such as scalar products ($fg$), vector products ($\mathbf{u}\times \mathbf{v}$), and combinations of the two ($f\mathbf{u}$).

Gradient of a Scalar Product

The gradient of a scalar product $fg$ is essentially an extension of the product rule for differentiation:

$[\nabla (fg){]}_i=\frac{\partial }{\partial x_i}(fg)=f\frac{\partial g}{\partial x_i}+g\frac{\partial f}{\partial x_i}=[f\nabla g+g\nabla f{]}_i$

Hence, $\boxed{\nabla (fg)=f\nabla g+g\nabla f}$.

Divergence of a Scalar Vector Product

Applying the product rule to the divergence of $f\mathbf{u}$:

$\nabla \cdot (f\mathbf{u})=\frac{\partial }{\partial x_i}(f{u}_i)=\frac{\partial f}{\partial x_i}{u}_i+f\frac{\partial u_i}{\partial x_i}$

Hence, $\boxed{\nabla \cdot (f\mathbf{u})=(\nabla f)\cdot \mathbf{u}+f(\nabla \cdot \mathbf{u})}$.

Curl of a Scalar Vector Product

Using the Alternating Tensor definition of the cross product:

$[\nabla \times (f\mathbf{u}){]}_i={ϵ}_{ijk}\frac{\partial }{\partial x_j}(f{u}_k)={ϵ}_{ijk}\frac{\partial f}{\partial x_j}{u}_k+f{ϵ}_{ijk}\frac{\partial u_k}{\partial x_j}$

$={ϵ}_{ijk}(\nabla f{)}_j{u}_k+f\left({ϵ}_{ijk}\frac{\partial u_k}{\partial x_j}\right)$

Hence, $\boxed{[\nabla \times (f\mathbf{u}){]}_i=[\nabla f\times \mathbf{u}+f(\nabla \times \mathbf{u}){]}_i}$.

Divergence of a Vector Product

For the divergence of $\mathbf{u}\times \mathbf{v}$:

$\nabla \cdot (\mathbf{u}\times \mathbf{v})=\frac{\partial }{\partial x_i}({ϵ}_{ijk}{u}_j{v}_k)={ϵ}_{ijk}\frac{\partial u_j}{\partial x_i}{v}_k+{ϵ}_{ijk}{u}_j\frac{\partial v_k}{\partial x_i}$

$=\left({ϵ}_{kij}\frac{\partial u_j}{\partial x_i}\right)v_k-\left({ϵ}_{jik}\frac{\partial v_k}{\partial x_i}\right)u_j$

Hence, $\boxed{\nabla \cdot (\mathbf{u}\times \mathbf{v})=(\nabla \times \mathbf{u})\cdot \mathbf{v}-(\nabla \times \mathbf{v})\cdot \mathbf{u}}$.

Curl of a Vector Product

The curl of a vector product $\mathbf{u}\times \mathbf{v}$ is slightly more complex. Expanding using the Kronecker Delta substitution ${ϵ}_{ijk}{ϵ}_{k\ell m}={\delta }_{i\ell }{\delta }_{jm}-{\delta }_{im}{\delta }_{j\ell }$:

$[\nabla \times (\mathbf{u}\times \mathbf{v}){]}_i={ϵ}_{ijk}\frac{\partial }{\partial x_j}({ϵ}_{k\ell m}{u}_{\ell }{v}_m)=({\delta }_{i\ell }{\delta }_{jm}-{\delta }_{im}{\delta }_{j\ell })\frac{\partial }{\partial x_j}(u_{\ell }{v}_m)$

$=\frac{\partial }{\partial x_j}(u_i{v}_j)-\frac{\partial }{\partial x_j}(u_j{v}_i)$

$=u_i\frac{\partial v_j}{\partial x_j}+v_j\frac{\partial u_i}{\partial x_j}-u_j\frac{\partial v_i}{\partial x_j}-v_i\frac{\partial u_j}{\partial x_j}$

To simplify this, we define a new operator: $\mathbf{u}\cdot \nabla =u_j\frac{\partial }{\partial x_j}$. Substituting this back in gives the final identity:

$\boxed{[\nabla \times (\mathbf{u}\times \mathbf{v}){]}_i=[\mathbf{u}(\nabla \cdot \mathbf{v})+(\mathbf{v}\cdot \nabla )\mathbf{u}-(\mathbf{u}\cdot \nabla )\mathbf{v}-\mathbf{v}(\nabla \cdot \mathbf{u}){]}_i}$.

Coordinate Systems

Coordinate systems are defined by a set of basis vectors (${\mathbf{v}}_1,{\mathbf{v}}_2,{\mathbf{v}}_3$) that must satisfy two conditions:

1. Linearly Independent: ${\lambda }_1{\mathbf{v}}_1+{\lambda }_2{\mathbf{v}}_2+{\lambda }_3{\mathbf{v}}_3=0$ is only true when ${\lambda }_1={\lambda }_2={\lambda }_3=0$.
2. Span the Space: Every vector $\mathbf{u}$ can be written as a linear combination $\mathbf{u}={\mu }_1{\mathbf{v}}_1+{\mu }_2{\mathbf{v}}_2+{\mu }_3{\mathbf{v}}_3$.

A coordinate system is orthogonal if its basis vectors intersect at $90^{\circ }$ angles (${\mathbf{v}}_i\cdot {\mathbf{v}}_j=0$ for $i\ne j$).

It is orthonormal if it is orthogonal and its basis vectors have a magnitude of 1 ($|{\mathbf{v}}_i|=1$). A standard example is the Cartesian coordinate system: $\mathbf{i}=(1,0,0),\mathbf{j}=(0,1,0),\mathbf{k}=(0,0,1)$.

Generalised Coordinate Systems

Generalised coordinate systems do not necessarily have orthogonal bases. For example, ${\mathbf{v}}_1=(1,0,0),{\mathbf{v}}_2=(2,1,0),{\mathbf{v}}_3=(0,0,1)$ are linearly independent and span the space, but they are not orthogonal since ${\mathbf{v}}_1\cdot {\mathbf{v}}_2=2\ne 0$.

2D Local Coordinate Transforms

If we define a 2D coordinate system by the plane $(x_1,x_2)$, we can rotate these axes by an angle $\theta$ to obtain a new coordinate system $(x_1^{\prime },x_2^{\prime })$. Any point $P$ transforms via:

$x_1^{\prime }=x_1\cos \theta +x_2\sin \theta$

$x_2^{\prime }=x_2\cos \theta -x_1\sin \theta$

In matrix form, this defines the Rotation Matrix $L$:

$$\left[\begin{array}{c}{x}_1^{\prime }\\ x_2^{\prime }\end{array}\right]=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]$$

Using suffix notation, this transformation is written compactly as $\boxed{x_i^{\prime }=L_{ij}{x}_j}$.

Properties of the Rotation Matrix

The rotation matrix $L$ has several important properties:

• Inverse is the Transpose: $L^{-1}=L^T$. This is because the inverse operation is simply a rotation by $-\theta$.
• Orthogonality: Since $L^T=L^{-1}$, we have $L{L}^T=I$ and $L^{T}L=I$. In suffix notation: $L_{ij}{L}_{kj}={\delta }_{ik}$ and $L_{ji}{L}_{jk}={\delta }_{ik}$.
• Determinant is 1: $|L|={\cos }^2\theta +{\sin }^2\theta =1$.

Because of these properties, we can easily find the inverse transformation. Multiplying $x_i^{\prime }=L_{ij}{x}_j$ by $L_{ik}$ yields:

$L_{ik}{x}_i^{\prime }=L_{ik}{L}_{ij}{x}_j={\delta }_{kj}{x}_j=x_k$

Hence, the inverse transformation is $\boxed{x_i=L_{ji}{x}_j^{\prime }}$.

---

Pre-Lecture Notes from University Notes

• Recap of combinations of grad, div, and curl on vector and scalar fields (div grad, curl grad, grad div, div curl, curl curl).
• Application of differential operators to products of functions using suffix notation:
  • Gradient of a scalar product: $\nabla (fg)=f\nabla g+g\nabla f$.
  • Divergence of a scalar vector product: $\nabla \cdot (f\mathbf{u})=(\nabla f)\cdot \mathbf{u}+f(\nabla \cdot \mathbf{u})$.
  • Curl of a scalar vector product: $[\nabla \times (f\mathbf{u}){]}_i=[\nabla f\times \mathbf{u}+f(\nabla \times \mathbf{u}){]}_i$.
  • Divergence of a vector product: $\nabla \cdot (\mathbf{u}\times \mathbf{v})=(\nabla \times \mathbf{u})\cdot \mathbf{v}-(\nabla \times \mathbf{v})\cdot \mathbf{u}$.
  • Curl of a vector product: requires expanding with Kronecker delta and defining the operator $\mathbf{u}\cdot \nabla =u_j\frac{\partial }{\partial x_j}$.
  • Gradient of a dot product (from practical questions): $\nabla (\mathbf{u}\cdot \mathbf{v})=\mathbf{u}\times (\nabla \times \mathbf{v})+\mathbf{v}\times (\nabla \times \mathbf{u})+(\mathbf{u}\cdot \nabla )\mathbf{v}+(\mathbf{v}\cdot \nabla )\mathbf{u}$.
• Introduction to Chapter 3: Local Coordinate Transforms.
  • Coordinate systems require basis vectors that are linearly independent and span the space.
  • Definitions of orthogonal (intersect at 90 degrees) and orthonormal (orthogonal and magnitude of 1) coordinate systems.
  • Cartesian system is an example of an orthonormal system.
  • Generalised coordinate systems do not need to be orthogonal.
• 2D coordinate system rotations:
  • Rotating axes by angle $\theta$ generates a transformation matrix $L$.
  • The new coordinates are defined by $x_i^{\prime }=L_{ij}{x}_j$.
  • Properties of rotation matrix $L$: $L^{-1}=L^T$, $L_{ij}{L}_{kj}={\delta }_{ik}$, and $|L|=1$.
  • The inverse transformation can be easily found using the transpose: $x_i=L_{ji}{x}_j^{\prime }$.
• Next lecture will expand rotating coordinate systems into 3D space.