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