Key Equations - Calculus Volume 3

Vector field in $ℝ^{2}$	$F (x, y) = 〈 P (x, y), Q (x, y) 〉$ or $F (x, y) = P (x, y) i + Q (x, y) j$
Vector field in $ℝ^{3}$	$F (x, y, z) = 〈 P (x, y, z), Q (x, y, z), R (x, y, z) 〉$ or $F (x, y, z) = P (x, y, z) i + Q (x, y, z) j + R (x, y, z) k$

Calculating a scalar line integral	$\int_{C} f (x, y, z) d s = \int_{a}^{b} f (r (t)) \sqrt{{(x^{'} (t))}^{2} + {(y^{'} (t))}^{2} + {(z^{'} (t))}^{2}} d t$
Calculating a vector line integral	$\int_{C} F \cdot d r = \int_{C} F \cdot T d s = \int_{a}^{b} F (r (t)) \cdot r^{'} (t) d t$ or $\int_{C} P d x + Q d y + R d z = \int_{a}^{b} (P (r (t)) \frac{d x}{d t} + Q (r (t)) \frac{d y}{d t} + R (r (t)) \frac{d z}{d t}) d t$
Calculating flux	$\int_{C} F \cdot \frac{n (t)}{‖ n (t) ‖} d s = \int_{a}^{b} F (r (t)) \cdot n (t) d t$

Fundamental Theorem for Line Integrals	$\int_{C} ∇ f \cdot d r = f (r (b)) - f (r (a))$
Circulation of a conservative field over curve C that encloses a simply connected region	$\int_{C} ∇ f \cdot d r = 0$

Green’s theorem, circulation form	$\int_{C} P d x + Q d y = \iint_{D} Q_{x} - P_{y} d A,$ where C is the boundary of D
Green’s theorem, flux form	$\int_{C} F \cdot N d s = \iint_{D} P_{x} + Q_{y} d A$
Green’s theorem, extended version	$\int_{\partial D} F \cdot d r = \iint_{D} Q_{x} - P_{y} d A$

Curl	$\nabla \times F = (R_{y} - Q_{z}) i + (P_{z} - R_{x}) j + (Q_{x} - P_{y}) k$
Divergence	$\nabla \cdot F = P_{x} + Q_{y} + R_{z}$
Divergence of curl is zero	$\nabla \cdot (\nabla \times F) = 0$
Curl of a gradient is the zero vector	$\nabla \times (∇ f) = 0$

Scalar surface integral	$\int \int_{S} f (x, y, z) d S = \int \int_{D} f (r (u, v)) \| \| t_{u} \times t_{v} \| \| d A$
Flux integral	$\iint_{S} F \cdot N d S = \iint_{S} F \cdot d S = \iint_{D} F (r (u, v)) \cdot (t_{u} \times t_{v}) d A$

Stokes’ theorem

\int_{C} F \cdot d r = \iint_{S} curl F \cdot d S

Divergence theorem

∭_{E} div F d V = \iint_{S} F \cdot d S