Curves, Velocity and Arc Length in Space: every key term you need (+ practice quiz)
25 flashcard terms for Multivariable Calculus Topic 2, written to match the course framework. Study them here, then drill them as interactive flashcards, or test yourself with the 15-question quiz — free, no account needed.
A function whose input is a real parameter and whose output is a vector, equivalent to a triple of ordinary component functions sharing one domain.
Space curve
The image traced in three dimensions by a continuous vector-valued function. The curve is a set of points, while the function that traces it is a parametrisation carrying extra timing information.
Limit of a vector function
Taken componentwise: the limit exists exactly when each component limit exists, and the limiting vector is assembled from those component limits.
Continuity of a vector function
Holds at a parameter value when each component function is continuous there, so the traced point moves without jumping.
Derivative of a vector function
The limit of a difference quotient of vectors, computed componentwise. Geometrically it points along the curve in the direction of increasing parameter.
Tangent vector
The nonzero derivative of a parametrisation at a point, spanning the tangent line to the curve there. It is undefined as a direction wherever the derivative vanishes.
Smooth curve
A parametrised curve whose derivative is continuous and never the zero vector on the interval. Smoothness is what rules out corners and cusps in the trace.
Velocity vector
The derivative of a position function with respect to time. Its direction is the instantaneous heading and its magnitude is the instantaneous speed.
Speed
The magnitude of the velocity vector, a nonnegative scalar. It measures how fast the parameter sweeps out length, not which way the motion goes.
Acceleration vector
The second derivative of position with respect to time. It generally has components both along and perpendicular to the direction of travel.
Tangential component of acceleration
The part of acceleration parallel to the velocity, equal to the rate of change of speed. It is zero exactly when speed is momentarily constant.
Normal component of acceleration
The part of acceleration perpendicular to the velocity, responsible for turning. It equals speed squared times curvature and vanishes only on a straight path.
Product rules for vector derivatives
Differentiation distributes across dot and cross products in the familiar pattern, but the cross-product version must preserve factor order because that product is anticommutative.
Derivative of a constant-length vector function
Always orthogonal to the function itself, since differentiating the constant dot product of the function with itself gives zero.
Integral of a vector function
Computed componentwise, producing a vector antiderivative plus an arbitrary constant vector. Recovering position from acceleration needs two such integrations and two vector initial conditions.
Arc length of a space curve
The integral of the speed over the parameter interval. For a smooth curve the value is the same for every parametrisation that traverses it once.
Arc length function
The length accumulated from a starting parameter up to a variable one. Its derivative is the speed, which is positive on a smooth curve, so the function is strictly increasing and invertible.
Arc length parametrisation
A reparametrisation using distance travelled as the parameter, so the speed is identically one. It strips timing information and leaves pure geometry.
Unit tangent vector
The velocity divided by the speed, written T. It exists wherever the curve is smooth and encodes the direction of travel only.
Principal unit normal vector
The derivative of the unit tangent divided by its own magnitude, written N. It is orthogonal to T and always points toward the concave side of the curve.
Binormal vector
The cross product of the unit tangent with the principal unit normal, completing a right-handed frame that moves along the curve.
Curvature
The magnitude of the rate of change of the unit tangent with respect to arc length, measuring how sharply a curve bends. It is zero on straight lines and constant on circles.
Osculating plane
The plane through a curve point spanned by the unit tangent and the principal unit normal. It is the plane the curve most nearly lies in near that point.
Torsion
The rate at which the binormal vector turns with respect to arc length, quantifying how a curve twists out of its osculating plane. Plane curves have zero torsion.
Radius of curvature
The reciprocal of the curvature at a point, equal to the radius of the circle that best fits the curve there. It becomes infinite where the curve is momentarily straight.