- #1

Odious Suspect

- 43

- 0

$$\nabla \times \mathfrak{v} = \lim_{\Delta \tau \to 0} \frac{1}{\Delta \tau }\oint d\mathfrak{S}\times \mathfrak{v}$$

I tried to demonstrate this, and neglected to place the surface element before the vector. My result seems to show that Joos's equation should have a negative sign on one side or the other.

My development assumes a finite cube centered on the origin with dimensions ##dx dy dz = d\tau## sufficiently small that deviations from the value of ##\mathfrak{v}(0,0,0)## are approximately linear. I sum the cross products of the field vector evaluated at the center of opposing faces with the surface element representing that face. Fiddle around with things, and end up with the curl.

Can someone tell me if I am doing something wrong here, and if so, what that something is?

$$\oint \mathfrak{v}\times d\mathfrak{S} \approx \left(\mathfrak{v}\left(\frac{dx}{2},0,0\right)\times \hat{\mathfrak{i}}+\mathfrak{v}\left(-\frac{dx}{2},0,0\right)\times \left(-\hat{\mathfrak{i}}\right)\right)dy dz

+\left(\mathfrak{v}\left(0,\frac{dy}{2},0\right)\times \hat{\mathfrak{j}}+\mathfrak{v}\left(0,-\frac{dy}{2},0\right)\times \left(-\hat{\mathfrak{j}}\right)\right)dx dz

+\left(\mathfrak{v}\left(0,0,\frac{dz}{2}\right)\times \hat{\mathfrak{k}}+\mathfrak{v}\left(0,0,-\frac{dz}{2}\right)\times \left(-\hat{\mathfrak{k}}\right)\right)dx dy$$

$$= \left(\mathfrak{v}\left(\frac{dx}{2},0,0\right)-\mathfrak{v}\left(-\frac{dx}{2},0,0\right)\right)\times \hat{\mathfrak{i}}dy dz

+\left(\mathfrak{v}\left(0,\frac{dy}{2},0\right)-\mathfrak{v}\left(0,-\frac{dy}{2},0\right)\right)\times \hat{\mathfrak{j}}dx dz

+\left(\mathfrak{v}\left(0,0,\frac{dz}{2}\right)-\mathfrak{v}\left(0,0,-\frac{dz}{2}\right)\right)\times \hat{\mathfrak{k}}dx dy $$

$$\approx \left(\left(\mathfrak{v}(0,0,0)+\frac{\partial }{\partial x}\mathfrak{v}(0,0,0)\frac{dx}{2}\right)-\left(\mathfrak{v}(0,0,0)-\frac{\partial }{\partial x}\mathfrak{v}(0,0,0)\frac{dx}{2}\right)\right)\times \hat{\mathfrak{i}}dydz

+\left(\left(\mathfrak{v}(0,0,0)+\frac{\partial }{\partial y}\mathfrak{v}(0,0,0)\frac{dy}{2}\right)-\left(\mathfrak{v}(0,0,0)-\frac{\partial }{\partial y}\mathfrak{v}(0,0,0)\frac{dy}{2}\right)\right)\times \hat{\mathfrak{j}}dzdx

+\left(\left(\mathfrak{v}(0,0,0)+\frac{\partial }{\partial z}\mathfrak{v}(0,0,0)\frac{dz}{2}\right)-\left(\mathfrak{v}(0,0,0)-\frac{\partial }{\partial z}\mathfrak{v}(0,0,0)\frac{dz}{2}\right)\right)\times \hat{\mathfrak{k}}dxdy$$

$$=\left(\frac{\partial }{\partial x}\mathfrak{v}(0,0,0)dx\right)\times \hat{\mathfrak{i}}dydz

+\left(\frac{\partial }{\partial x}\mathfrak{v}(0,0,0)dy\right)\times \hat{\mathfrak{j}}dzdx

+\left(\frac{\partial }{\partial x}\mathfrak{v}(0,0,0)dz\right)\times \hat{\mathfrak{k}}dxdy$$

$$=\left(\left(\frac{\partial }{\partial x}v_x\hat{\mathfrak{i}}+\frac{\partial }{\partial y}v_x\hat{\mathfrak{j}}+\frac{\partial }{\partial z}v_x\hat{\mathfrak{k}}\right)\times \hat{\mathfrak{i}}+\left(\frac{\partial }{\partial x}v_y\hat{\mathfrak{i}}+\frac{\partial }{\partial y}v_y\hat{\mathfrak{j}}+\frac{\partial }{\partial z}v_y\hat{\mathfrak{k}}\right)\times \hat{\mathfrak{j}}+\left(\frac{\partial }{\partial x}v_z\hat{\mathfrak{i}}+\frac{\partial }{\partial y}v_z\hat{\mathfrak{j}}+\frac{\partial }{\partial z}v_z\hat{\mathfrak{k}}\right)\times \hat{\mathfrak{k}}\right)d\tau$$

$$=\left(-\frac{\partial }{\partial y}v_x\hat{\mathfrak{k}}+\frac{\partial }{\partial z}v_x\hat{\mathfrak{j}}+\frac{\partial }{\partial x}v_y\hat{\mathfrak{k}}-\frac{\partial }{\partial z}v_y\hat{\mathfrak{i}}-\frac{\partial }{\partial x}v_z\hat{\mathfrak{j}}+\frac{\partial }{\partial y}v_z\hat{\mathfrak{i}}\right)d\tau$$

$$=\left(\left(\frac{\partial }{\partial y}v_z-\frac{\partial }{\partial z}v_y\right)\hat{\mathfrak{i}}+\left(\frac{\partial }{\partial z}v_x-\frac{\partial }{\partial x}v_z\right)\hat{\mathfrak{j}}+\left(\frac{\partial }{\partial x}v_y-\frac{\partial }{\partial y}v_x\right)\hat{\mathfrak{k}}\right)d\tau$$

$$\lim_{d\tau \to 0} \frac{1}{d\tau }\oint \mathfrak{v}\times d\mathfrak{S}=\left(\frac{\partial }{\partial y}v_z-\frac{\partial }{\partial z}v_y\right)\hat{\mathfrak{i}}+\left(\frac{\partial }{\partial z}v_x-\frac{\partial }{\partial x}v_z\right)\hat{\mathfrak{j}}+\left(\frac{\partial }{\partial x}v_y-\frac{\partial }{\partial y}v_x\right)\hat{\mathfrak{k}}$$