Gauss’s Law states that:
∫ ∫ S E ⋅ n ^ d S = q ϵ 0 \int \int _S \mathbf{E} \cdot \hat{n} dS = \frac{q}{\epsilon_0} ∫ ∫ S E ⋅ n ^ d S = ϵ 0 q where q q q
What is the n ^ \hat{n} n ^ n ^ \hat{n} n ^ u a n d v \mathbf{u} and \mathbf{v} u an d v
n ^ = u × v ∣ u × v ∣ . \hat{n} = \frac{\mathbf{u} \times \mathbf{v}}{|\mathbf{u} \times \mathbf{v}|}. n ^ = ∣ u × v ∣ u × v . (Check the book for more detailed derivation)
First:
u = [ i ^ + ∂ f ∂ x k ^ ] u x \mathbf{u} = \left[\hat{i} + \frac{\partial f}{\partial x} \hat{k} \right] u_x u = [ i ^ + ∂ x ∂ f k ^ ] u x Second:
v = [ j ^ + ∂ f ∂ y k ^ ] v y \mathbf{v} = \left[\hat{j} + \frac{\partial f}{\partial y} \hat{k} \right] v_y v = [ j ^ + ∂ y ∂ f k ^ ] v y Then:
u × v = ( u x 0 u x ∂ f ∂ x ) × ( 0 v y v y ∂ f ∂ y ) \mathbf{u} \times \mathbf{v} = \begin{pmatrix}
u_x \\
0 \\
u_x \frac{\partial f}{\partial x}
\end{pmatrix}
\times
\begin{pmatrix}
0 \\
v_y \\
v_y \frac{\partial f}{\partial y}
\end{pmatrix} u × v = u x 0 u x ∂ x ∂ f × 0 v y v y ∂ y ∂ f