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The integral of the product of two functions $u$ and $dv$ is $\int u dv = uv - \int v du$. It's used to integrate products of functions that are not easily integrable in their current form.

The curl of a vector field $\vec{F} = \langle P, Q, R \rangle$ is $\nabla \times \vec{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right)\vec{i} + \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right)\vec{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right)\vec{k}$. It measures the rotation of the field at a point.

The derivative of $x^n$ where $n$ is a constant is $nx^{n-1}$. It's one of the most basic rules used to differentiate powers of $x$.

A method to find the local maxima and minima of a function subject to equality constraints. If $f(x,y,...)$ is the function and $g(x,y,...)=0$ is the constraint, then the extrema occur at points where $\nabla f = \lambda \nabla g$ where $\lambda$ is the Lagrange multiplier.

The Taylor series for a function $f$ around $a$ is $\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$. It's used to approximate functions using polynomials.

Relates a surface integral over a surface $S$ in a vector field $\vec{F}$ to a line integral around the boundary curve $C$ of $S$: $\oint_C \vec{F} \cdot d\vec{r} = \int_S (\nabla \times \vec{F}) \cdot d\vec{S}$. It's used to convert between line integrals and surface integrals.

The derivative of a composite function $f(g(x))$ is $f'(g(x))\cdot g'(x)$. It's used when a function is nested inside another function.

Used when a function $y$ is defined implicitly as a function of $x$, the derivative $\frac{dy}{dx}$ is found by differentiating both sides of the equation with respect to $x$ and solving for $\frac{dy}{dx}$. It's used when functions can't be easily solved for a variable.

If the limit $\lim_{x\to c} \frac{f(x)}{g(x)}$ results in indeterminate forms like $0/0$ or $\infty/\infty$, then under certain conditions, it equals $\lim_{x\to c} \frac{f'(x)}{g'(x)}$. It's used to evaluate limits involving indeterminate forms.

If $C$ is a positively oriented, piecewise-smooth, simple closed curve in a plane, and $D$ is the region bounded by $C$, then $\oint_C (P dx + Q dy) = \iint_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) dA$. It's used to relate a line integral around a simple closed curve to a double integral over the plane region it encloses.

If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, there exists some $c$ in $(a,b)$ such that $f'(c) = \frac{f(b)-f(a)}{b-a}$. It's used to show that a function takes on its mean slope at least once.

If a continuous function $f$ has values $f(a)$ and $f(b)$ of opposite signs on an interval $[a,b]$, then there exists at least one $c \in (a,b)$ such that $f(c)=0$. It's used to guarantee the existence of roots in an interval.

If $u$ and $v$ are functions of $x$, then the derivative of their product $u(x)v(x)$ is given by $u'v + uv'$. It's used to differentiate the product of two functions.

The derivative of $f$ at $x=a$ is given by $\lim_{h\to 0} \frac{f(a+h) - f(a)}{h}$. It's used as the foundational definition of a derivative, representing the instantaneous rate of change of the function.

The integral of $x^n$ where $n \neq -1$ is $\frac{x^{n+1}}{n+1} + C$ where $C$ is the constant of integration. It's used to find antiderivatives of power functions.

A method for expressing a rational function as the sum of simpler fractions. It's used to simplify the integration of rational functions.

For non-negative real numbers $a_1, a_2, ..., a_n$, the inequality states $\sqrt[n]{a_1 a_2 ... a_n} \leq \frac{a_1 + a_2 + ... + a_n}{n}$ with equality if and only if all $a_i$ are equal. It's used to relate the mean of values in their product form to their sum form.

If $u$ and $v$ are functions of $x$, then the derivative of their quotient $\frac{u(x)}{v(x)}$ is given by $\frac{u'v - uv'}{v^2}$. It's used to differentiate ratios of functions.

For a gradient field $F=\nabla f$, the line integral over a curve $C$ from point $a$ to point $b$ is $\int_C F\cdot dr = f(b) - f(a)$. It's used to evaluate line integrals in conservative fields without parameterizing the curve.

For an electric field $\vec{E}$ and a closed surface $S$, the law states $\oint_S \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\varepsilon_0}$. It relates the electric flux through a closed surface to the charge enclosed by the surface.

Provides a connection between differentiation and integration: if $f$ is continuous on $[a,b]$ and $F$ is an antiderivative of $f$, then $\int_a^b f(x)dx = F(b) - F(a)$. It's used to evaluate definite integrals.

For a function $f$, if $f'(c)=0$ and $f''(c)>0$, $c$ is a local minimum. If $f''(c)<0$, $c$ is a local maximum. It's used to determine concavity and whether a critical point is a max or min.

For a vector field $\vec{F}$ and a solid region $V$ with boundary surface $S$, the theorem states $\int_V (\nabla \cdot \vec{F}) dV = \oint_S \vec{F} \cdot d\vec{S}$. It's used to relate the flow of a field through a surface to the behavior of the field inside the volume.