0.2 Cartesian vectors and tensors: their calculus  (Page 9/9)

Considering $\mathbf{I}$ and $\mathbf{J}$ as representing currents, the first equation expresses that the amount which flows out of a small region is equal to the amount which flows in. The second equation expresses that the total flow from a portion of a conducting surface into space and along the surface is zero. The third equation expresses that the flow from one sub-region across a curve on a conducting surface is equal to the flow into the adjacent sub-region. These three equations thus express that $\mathbf{I}$ and $\mathbf{J}$ , considered as space and surface currents, represent a flow of something which is conserved. For $\mathbf{I}$ and $\mathbf{J}$ to have this property the above discussion shows it is necessary and sufficient that A be everywhere solenoidal. In hydrodynamics, the field $\mathbf{I}$ corresponds to vorticity and it clearly is solenoidal because it is the curl of velocity.

Vector and scalar potential

The previous section showed that a vector field can be determined from the divergence and curl of the vector field and the values on surfaces of discontinuities and bounding surfaces. The integral equations are useful for developing analytical solutions for simple systems. However, in hydrodynamics the vorticity is generally an unknown quantity. Thus it is useful to express the potentials as differential equations that are solved simultaneously for the potentials and vorticity. The differential equations are derived by substituting the potentials into the expressions for the divergence and curl.

In two-dimensional vector fields the vector potential $\mathbf{A}$ and the vector $\mathbf{I}$ has a nonzero component only in the third direction. In hydrodynamics the nonzero component of the vector potential is the stream function $4\pi \mathbf{I}$ corresponds to the vorticity, which has only one nonzero component.

Assignment 3.1: particle velocity and acceleration

1. Show that its acceleration is perpendicular to its velocity.
2. Show that the acceleration has a radial (from the center of the sphere) component $a_{\rho }=-v^2/R$ .
3. Let the magnitude of the angular velocity be $\omega$ . Express the centrifugal acceleration (direction perpendicular to the axis of rotation), $a$ in terms of $R$ , $\omega$ and the angle of particle from the axis of rotation.

Assignment 3.2: differential area and volume

1. Express the differential of area in term of the differential of the surface coordinates for a spherical surface using the spherical polar coordinate system.
2. Obtain the differential volume elements in cylindrical and spherical polars by the Jacobian and check with a simple geometrical picture.

Assignment 3.3: differential operators

1. Derive the expression for the Laplacian of a scalar in Cartesian coordinates from the definition of the gradient and divergence.
2. Prove that: $\nabla •\left(\varphi \mathbf{a}\right)=\nabla \varphi •\mathbf{a}+\varphi (\nabla •\mathbf{a})$
3. Let $\mathbf{x}$ be the Cartesian coordinates of points in space and $r=\left|\mathbf{x}\right|$ . Calculate the divergence and curl of $\mathbf{x}$ and the gradient and Laplacian of $r$ and $1/r$ . Note any singularities.
4. Prove the identities involving the curl operator.
5. Suppose a rigid body has the velocity field $\mathbf{v}={\mathbf{v}}_{\left(t\right)}+\omega \times \left(\mathbf{x}-{\mathbf{x}}_o\right)$ . Show that the curl of this velocity field is $\nabla \times \mathbf{v}=2\omega$ .