Topics · 12 formulas

Parametric, Polar & Vectors

Parametric curves, polar coordinates, and vector operations.

Parametric Derivative (dy/dx)

\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{y'(t)}{x'(t)}

The slope of a parametric curve x = x(t), y = y(t) is the ratio of the derivatives with respect to the parameter t.

Conditions: x'(t) ≠ 0.

Parametric Curves

Parametric Arc Length

L = \int_\alpha^\beta \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}\, dt

The length of a parametric curve from t = α to t = β.

Parametric Curves

Parametric Area

A = \int_\alpha^\beta y(t)\, x'(t)\, dt

The area under a parametric curve. More precisely, the signed area between the curve and the x-axis.

Parametric Curves

Polar to Cartesian

x = r\cos\theta, \quad y = r\sin\theta, \quad r^2 = x^2 + y^2, \quad \tan\theta = y/x

Conversion formulas between polar coordinates (r, θ) and Cartesian coordinates (x, y).

Polar Coordinates

Polar Area

A = \frac{1}{2}\int_\alpha^\beta [r(\theta)]^2 \, d\theta

The area enclosed by a polar curve r = f(θ) from θ = α to θ = β.

Polar Coordinates

Polar Arc Length

L = \int_\alpha^\beta \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2}\, d\theta

The arc length of a polar curve r = f(θ) from θ = α to θ = β.

Polar Coordinates

Polar Curve Slope

\frac{dy}{dx} = \frac{r'\sin\theta + r\cos\theta}{r'\cos\theta - r\sin\theta}

The slope of a polar curve at a given angle θ. Derived from x = r cos θ, y = r sin θ using the parametric derivative formula.

Polar Coordinates

Vector Magnitude

|\vec{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2}

The magnitude (length) of a vector v = ⟨v₁, v₂, v₃⟩.

Vector Operations

Dot Product

\vec{u} \cdot \vec{v} = u_1 v_1 + u_2 v_2 + u_3 v_3 = |\vec{u}||\vec{v}|\cos\theta

The dot product is a scalar equal to the sum of component products. It also equals the product of magnitudes times the cosine of the angle between them.

Vector Operations

Cross Product

\vec{u} \times \vec{v} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{vmatrix}

The cross product produces a vector perpendicular to both u and v. Its magnitude equals |u||v|sin θ (the area of the parallelogram formed by u and v).

Vector Operations

Vector Projection

\text{proj}_{\vec{v}} \vec{u} = \frac{\vec{u} \cdot \vec{v}}{|\vec{v}|^2} \vec{v}

The projection of u onto v gives the component of u in the direction of v.

Vector Operations

Angle Between Vectors

\cos\theta = \frac{\vec{u} \cdot \vec{v}}{|\vec{u}||\vec{v}|}

The angle between two vectors is found using the dot product divided by the product of their magnitudes.

Vector Operations