# Surface Integrals

Physics I · Differential Equations and Vector Calculus · https://tryals.app/learn/physics-i/surface-integrals

## Surface Integrals & Parametrisation

A **surface integral** extends integration from curves to 2D surfaces in 3D space.

A **parametric surface** $S$ is defined by parameters $(u,v)$:

$\vec{r}(u,v) = x(u,v)\hat{i} + y(u,v)\hat{j} + z(u,v)\hat{k}$

The differential area vector is built from partial derivatives:

$d\vec{S} = \left(\frac{\partial\vec{r}}{\partial u}\times\frac{\partial\vec{r}}{\partial v}\right)du\,dv$

A **scalar surface integral** integrates a function $f$ over $S$:

$\iint_S f\,dS = \iint_D f(\vec{r}(u,v))\,\left\|\frac{\partial\vec{r}}{\partial u}\times\frac{\partial\vec{r}}{\partial v}\right\|\,du\,dv$

| Surface Type | Area Element $d\vec{S}$ |
| --- | --- |
| Parametric | $\left(\frac{\partial\vec{r}}{\partial u}\times\frac{\partial\vec{r}}{\partial v}\right)du\,dv$ |
| Graph $z = g(x,y)$ | $(-g_x\hat{i} - g_y\hat{j} + \hat{k})\,dx\,dy$ |

## Flux Integrals & Orientation

A **flux integral** measures the net flow of a vector field $\vec{F}$ through a surface $S$:

$\Phi = \iint_S \vec{F}\cdot d\vec{S} = \iint_S \vec{F}\cdot\hat{n}\,dS$

**Physics link**: Electric flux through a closed surface equals the enclosed charge via Gauss's law:

$\Phi_E = \oiint_S \vec{E}\cdot d\vec{S} = \frac{Q_{\text{enc}}}{\varepsilon_0}$

**Orientation** is critical: the choice of unit normal $\hat{n}$ (outward vs. inward for closed surfaces) determines the sign of the flux.

**Common pitfall**: Flux integrals are orientation-sensitive. Flipping the surface normal flips the sign of your final answer. Always fix the normal direction before you begin integrating.

## Practice questions

5 of this lesson's 13 practice questions, with answers. The full set is in the app.

### 1. Sort: is this a line integral or a surface integral?

**Answer:**

- Line integral: $\int_C F \cdot dr$, Work done by a force along a path, Circulation around a loop
- Surface integral: ∬_S F $\cdot dS$, Flux through a membrane, Electric flux through a Gaussian surface

**Why:** Line integrals are over curves (1D); surface integrals are over surfaces (2D).

Page: https://tryals.app/practice/physics-i/surface-integrals/sort-is-this-a-line-integral-or-a-surface-integral

### 2. Rain falls straight down. You hold a large flat board out in the storm. When does the board intercept the **most** water per second?

A. Tilted at 45°
B. Held vertically, edge-up to the rain
C. Orientation makes no difference for the same board
D. Held horizontally, face-up to the rain

**Answer:** D. Held horizontally, face-up to the rain

**Why:** Flux is maximal when the surface normal aligns with the flow ($\cos\theta = 1$) and zero when the surface is edge-on. Every surface integral $\iint \vec{F}\cdot d\vec{S}$ is this rain-catching computation, patch by patch, over a curved surface.

Page: https://tryals.app/practice/physics-i/surface-integrals/rain-falls-straight-down-you-hold-a-large-flat-board-out-in-the

### 3. A flux integral needs a chosen orientation for the surface, while a surface-area integral does not. Why does flux care about the side?

A. Because flux is only defined for closed surfaces
B. Because flux measures what passes THROUGH, and in differs from out
C. Because area integrals are always positive by convention
D. Because the orientation determines which parametrisation is valid

**Answer:** B. Because flux measures what passes THROUGH, and in differs from out

**Why:** Direction is part of the quantity. Gauss’s law depends on it entirely: a net outward flux means enclosed charge, and the sign is the whole content of the statement.

Page: https://tryals.app/practice/physics-i/surface-integrals/a-flux-integral-needs-a-chosen-orientation-for-the-surface-while-a

### 4. Complete the statement with the correct term.

**Answer:** To compute a surface integral, the surface is described by two parameters, traditionally called $u$ and **v**.

**Why:** A surface is two-dimensional, so two parameters $(u, v)$ are needed, just as one parameter $t$ suffices for a curve.

Page: https://tryals.app/practice/physics-i/surface-integrals/complete-the-statement-with-the-correct-term

### 5. Complete the following passage by choosing the correct terms from the word bank.

**Answer:** The magnitude |rᵤ $\times r$ᵥ| gives the local **area** of the surface element, and its direction gives the surface **normal**.

**Why:** The cross product of tangent vectors gives the normal with magnitude = area element.

Page: https://tryals.app/practice/physics-i/surface-integrals/complete-the-following-passage-by-choosing-the-correct-terms-from-the