Vector Field Flux Calculation Fundamentals
📌 The elementary flux ($\text{d}\Phi$) is defined as the scalar product of the vector field $\mathbf{V}(\mathbf{M})$ and the elementary surface vector $\text{d}\mathbf{S}$, where $\text{d}\mathbf{S} = \mathbf{n} \, \text{d}S$ ($\mathbf{n}$ is the unit vector normal to the surface).
📐 If $\alpha$ is the angle between $\mathbf{n}$ and $\mathbf{V}$, the flux is positive when $0 \leq \alpha < \pi/2$ and negative when $\pi/2 < \alpha \leq \pi$.
🌊 The total flux ($\Phi$) through a surface $S$ is the surface integral of the elementary flux: $\Phi = \iint_S \mathbf{V} \cdot \text{d}\mathbf{S} = \iint_S \mathbf{V} \cdot \mathbf{n} \, \text{d}S$.

Flux Calculation Through a Cube Surface
📐 The vector field for the exercise is defined as $\mathbf{V}(\mathbf{M}) = 4x Z \, \mathbf{e}_x + y^2 \, \mathbf{e}_y + YZ \, \mathbf{e}_z$ over a cube defined by $0 \leq X \leq 1$, $0 \leq Y \leq 1$, and $0 \leq Z \leq 1$.
📉 The flux through the face $X=0$ ($\mathbf{n} = -\mathbf{e}_x$) is zero ($\Phi_1 = 0$) because the component of $\mathbf{V}$ in the $\mathbf{e}_x$ direction ($4xZ$) becomes zero when $X=0$.
➕ The flux through the face $X=1$ ($\mathbf{n} = +\mathbf{e}_x$) is calculated as $\Phi_2 = \iint (4Z \, \mathbf{e}_x) \cdot (\mathbf{e}_x \, \text{d}y \, \text{d}Z) = \int_0^1 \int_0^1 4Z \, \text{d}y \, \text{d}Z$, resulting in $\Phi_2 = 2$.
🚫 The flux through the face $Y=0$ ($\mathbf{n} = -\mathbf{e}_y$) is zero ($\Phi_3 = 0$) because the $\mathbf{e}_y$ component of $\mathbf{V}$ ($y^2$) is zero when $Y=0$.
➕ The flux through the face $Y=1$ ($\mathbf{n} = +\mathbf{e}_y$) is calculated as $\Phi_4 = \iint (y^2 \, \mathbf{e}_y) \cdot (\mathbf{e}_y \, \text{d}x \, \text{d}Z) = \int_0^1 \int_0^1 y^2 \, \text{d}x \, \text{d}Z$, yielding $\Phi_4 = 1/3$. *(Correction noted in final summation: the calculation provided resulted in $\Phi_4 = 1$ based on the integration shown, $\int_0^1 1^2 \, \text{d}y \times \int_0^1 1 \, \text{d}x = 1$. Following the calculation in the video)*.
❓ The flux through the face $Z=0$ ($\mathbf{n} = -\mathbf{e}_z$) is zero ($\Phi_5 = 0$) because the $\mathbf{e}_z$ component of $\mathbf{V}$ ($YZ$) is zero when $Z=0$.
➕ The flux through the face $Z=1$ ($\mathbf{n} = +\mathbf{e}_z$) is calculated as $\Phi_6 = \iint (YZ \, \mathbf{e}_z) \cdot (\mathbf{e}_z \, \text{d}x \, \text{d}y) = \int_0^1 \int_0^1 Y \, \text{d}x \, \text{d}y$, resulting in $\Phi_6 = 1/2$.

Total Flux Result
🔗 The total flux $\Phi_{\text{total}}$ is the sum of the fluxes through all six faces: $\Phi_{\text{total}} = \Phi_1 + \Phi_2 + \Phi_3 + \Phi_4 + \Phi_5 + \Phi_6$.
🧮 Summing the calculated values: $\Phi_{\text{total}} = 0 + 2 + 0 + 1 + 0 + 1/2$. *(Using the video's calculated values for $\Phi_2=2$, $\Phi_4=1$, $\Phi_6=1/2$)*.
✅ The final total flux calculated is $\Phi_{\text{total}} = 3 + 1/2 = \mathbf{3.5}$ or $\mathbf{7/2}$. *(The video shows $2 - 1 + 1/2 = 1 + 1/2 = 3/2$, suggesting an error in transcribing $\Phi_4$ or a calculation mistake during summation)*.

Key Points & Insights
➡️ Calculate flux separately for each face by determining the outward normal vector ($\mathbf{n}$) and evaluating $\mathbf{V} \cdot \mathbf{n}$ on that surface.
➡️ Zero flux occurs when the component of the vector field normal to the surface is zero (e.g., when the field has no normal component, or when the field component responsible for flux is zero on the boundary plane).
➡️ For the face $X=1$, the flux was calculated as $2$, derived from $\int_0^1 4Z \, \text{d}Z \times \int_0^1 \text{d}Y = [2Z^2]_0^1 \times [Y]_0^1 = 2 \times 1 = 2$.
➡️ The total flux calculation in the video results in $\mathbf{3/2}$ based on $\Phi_2=2$, $\Phi_4=1$, and $\Phi_6=1/2$ (if $\Phi_4$ was $1$ and not $1/3$), but the final sum shown is $2 - 1 + 1/2 = 3/2$.

📸 Video summarized with SummaryTube.com on Dec 28, 2025, 22:59 UTC