Line Integrals and Boundary Orientation Rules

Vocabulary and conceptual flashcards covering line integral conventions, boundary curve orientations, and path decomposition cases based on lecture notes.

Tangential component of a vector field ($\vec{F}_{tan}$)

The component of a vector field $\vec{F}$ projected along the unit tangent vector $\vec{t}_p$, formulated as $\vec{F}_{tan} = (\vec{F} \cdot \vec{t}_p)\vec{t}_p$.

Surface to the left rule

A standard orientation rule stating that during the total traversal of a boundary in the counterclockwise direction ("sentido anti-horario"), the enclosed region or surface must always remain on the left-hand side.

Touching closed regions (Case 1)

A configuration where the boundary separates into two distinct closed curves or subregions ($\partial R_1 = \partial \Omega_1$ and $\partial R_2 = \partial \Omega_2$) meeting at a point $P_0(t_0)$, requiring $2$ separate parametrizations ($\Phi_1 = (dx_1, dy_1)$ and $\Phi_2 = (dx_2, dy_2)$) and evaluating the integral over each curve.

Curve with axis contact (Case 2)

A configuration where a loop makes contact with a straight axis or line; since integration occurs strictly along the boundary, only the closed curve itself is parameterized ($\Phi_1 = (dx, dy)$), yielding a total of $1$ path to evaluate.

Connected interior hole (Case 3)

A multiply connected region containing an inner hole bounded by an inner and an outer curve ($2$ curves in total), where the combined line integral is found via the subtraction of curve integrals: $I_{\text{ext}} - I_{\text{int}}$ (or the reverse depending on orientation).

Subtraction of curve integrals ($I_{\text{ext}} - I_{\text{int}}$)

The operation used to evaluate the total line integral over a region with an interior hole, calculated by taking the difference between the integral along the exterior boundary ($I_{\text{ext}}$) and the interior boundary ($I_{\text{int}}$).