Glossary

2.2 Derivatives and Integrals of Vector-Valued Functions

2 min read•july 25, 2024

are powerful tools for describing motion in space. They map scalar inputs to vector outputs, allowing us to represent position, velocity, and acceleration as they change over time.

Differentiation and integration of these functions follow familiar rules, but with vector components. This lets us analyze curves, find tangent lines, and solve motion problems using calculus techniques adapted for multidimensional space.

Vector-Valued Functions: Derivatives and Integrals

Differentiation of vector-valued functions

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  • Vector-valued functions map scalar inputs to vector outputs r(t)=⟨f(t),g(t),h(t)⟩\mathbf{r}(t) = \langle f(t), g(t), h(t) \rangle
    • Component functions f(t)f(t), g(t)g(t), and h(t)h(t) describe individual coordinate behavior
  • Derivative r′(t)=⟨f′(t),g′(t),h′(t)⟩\mathbf{r}'(t) = \langle f'(t), g'(t), h'(t) \rangle differentiates each component function
    • Geometrically represents indicating direction of motion at a point
  • Tangent vectors calculated using derivative provide instantaneous direction of curve
  • L(t)=r(t0)+tr′(t0)\mathbf{L}(t) = \mathbf{r}(t_0) + t\mathbf{r}'(t_0) uses point and direction to define line

Rules for vector function differentiation

  • Sum rule (u+v)′=u′+v′(\mathbf{u} + \mathbf{v})' = \mathbf{u}' + \mathbf{v}' differentiates vector components individually
  • Scalar multiple rule (cu)′=cu′(c\mathbf{u})' = c\mathbf{u}' applies constant to derivative
  • Product rules extend scalar concepts to vectors
    • ddt(u⋅v)=u′⋅v+u⋅v′\frac{d}{dt}(\mathbf{u} \cdot \mathbf{v}) = \mathbf{u}' \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{v}'
    • ddt(u×v)=u′×v+u×v′\frac{d}{dt}(\mathbf{u} \times \mathbf{v}) = \mathbf{u}' \times \mathbf{v} + \mathbf{u} \times \mathbf{v}'
  • ddtf(g(t))=g′(t)f′(g(t))\frac{d}{dt}\mathbf{f}(g(t)) = g'(t)\mathbf{f}'(g(t)) applies to composite vector functions

Integration of vector-valued functions

  • ∫r(t)dt=⟨∫f(t)dt,∫g(t)dt,∫h(t)dt⟩+C\int \mathbf{r}(t) dt = \langle \int f(t) dt, \int g(t) dt, \int h(t) dt \rangle + \mathbf{C} integrates each component
  • ∫abr(t)dt=⟨∫abf(t)dt,∫abg(t)dt,∫abh(t)dt⟩\int_a^b \mathbf{r}(t) dt = \langle \int_a^b f(t) dt, \int_a^b g(t) dt, \int_a^b h(t) dt \rangle evaluates over interval
  • Initial value problems solve for position function given initial conditions
    1. Use given initial velocity
    2. Integrate to find position function
    3. Apply initial position to determine constant of integration
  • ddt∫atr(s)ds=r(t)\frac{d}{dt} \int_a^t \mathbf{r}(s) ds = \mathbf{r}(t) extends to vector functions

Motion in Space

Position, velocity, and acceleration relationships

  • Position vector r(t)\mathbf{r}(t) describes object's location in space over time
  • Velocity vector v(t)=r′(t)\mathbf{v}(t) = \mathbf{r}'(t) represents rate of change of position
    • Tangent to path of motion, indicating instantaneous direction
  • Acceleration vector a(t)=v′(t)=r′′(t)\mathbf{a}(t) = \mathbf{v}'(t) = \mathbf{r}''(t) shows rate of change of velocity
  • Speed ∥v(t)∥\|\mathbf{v}(t)\| measures magnitude of velocity (scalar quantity)
  • s=∫ab∥r′(t)∥dts = \int_a^b \|\mathbf{r}'(t)\| dt calculates distance traveled along curve
  • T(t)=r′(t)∥r′(t)∥\mathbf{T}(t) = \frac{\mathbf{r}'(t)}{\|\mathbf{r}'(t)\|} provides normalized direction of motion
  • N(t)=T′(t)∥T′(t)∥\mathbf{N}(t) = \frac{\mathbf{T}'(t)}{\|\mathbf{T}'(t)\|} points perpendicular to motion
  • B(t)=T(t)×N(t)\mathbf{B}(t) = \mathbf{T}(t) \times \mathbf{N}(t) completes right-handed coordinate system

Key Terms to Review (16)

Arc Length: Arc length is the measure of the distance along a curve between two points. This concept is particularly relevant in the context of vector-valued functions, where it allows us to quantify the length of a path traced out by a function in multi-dimensional space. Understanding arc length aids in the analysis of motion, curvature, and the geometry of curves, all of which are fundamental aspects of calculus involving vectors.
Binormal vector: The binormal vector is a vector that is orthogonal to both the tangent and normal vectors of a space curve, forming part of the Frenet-Serret frame. It provides important information about the twisting of the curve in three-dimensional space. This vector is crucial for understanding the curvature and torsion of a curve, which are key concepts related to how curves behave and change direction.
Chain Rule: The chain rule is a fundamental concept in calculus that provides a way to compute the derivative of a composite function. It allows you to differentiate functions that are nested within one another by relating the rates of change of the outer function to the rates of change of the inner function. This is especially important when dealing with functions of several variables or vector-valued functions.
Cross product: The cross product is a mathematical operation on two vectors in three-dimensional space that produces a third vector which is orthogonal to both of the original vectors. This operation not only helps to find the direction and area of parallelograms defined by the two vectors, but it also plays a significant role in calculating torque and angular momentum in physics.
Curve length: Curve length is the total distance along a curved path between two points in space. This concept is fundamental in understanding how to measure complex shapes and paths, especially in the context of vector-valued functions, where curves are often represented parametrically. By determining the length of a curve, one can analyze its properties and behavior over a specified interval, which is essential when dealing with motion and trajectories in a multi-dimensional setting.
Definite integral: A definite integral is a mathematical concept that represents the accumulation of quantities, such as area under a curve, between two specific bounds. It is expressed as $$\int_{a}^{b} f(x) \, dx$$, where $$a$$ and $$b$$ are the limits of integration. This integral quantifies the total change of a function over a given interval, connecting seamlessly with various applications, including arc length and the analysis of vector-valued functions.
Derivative of a vector function: The derivative of a vector function is a mathematical concept that describes how a vector-valued function changes with respect to its input variable, often time or another parameter. This derivative captures the rate of change of each component of the vector function and is represented as a new vector. It plays a crucial role in understanding motion and trajectories in space, as it allows for the analysis of speed and direction at any point along a path.
Dot product: The dot product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. It measures the extent to which two vectors point in the same direction, providing insight into the angle between them and their geometric relationship. This operation is fundamental in various mathematical contexts, including geometry, physics, and computer science.
Fundamental Theorem of Calculus: The Fundamental Theorem of Calculus connects the concepts of differentiation and integration, showing that they are essentially inverse processes. It establishes a relationship between the definite integral of a function and its antiderivative, allowing for the computation of areas under curves and the evaluation of integrals using the antiderivative. This theorem is foundational for understanding various advanced concepts in calculus, including vector-valued functions and the application of line integrals in multivariable calculus.
Indefinite Integral: An indefinite integral represents a family of functions whose derivative is the integrand. It is expressed without specific limits of integration and is essentially the reverse process of differentiation, capturing all antiderivatives of a function. This concept is crucial for understanding how to work with vector-valued functions, as it allows for the accumulation of quantities and helps in solving problems involving motion and curves in higher dimensions.
Normal Vector: A normal vector is a vector that is perpendicular to a given surface or curve at a specific point. It provides crucial information about the orientation of surfaces in three-dimensional space and is essential for various applications such as calculating surface integrals, determining curvature, and analyzing geometric properties of curves and surfaces.
Tangent line equation: The tangent line equation represents a straight line that touches a curve at a specific point, indicating the direction of the curve at that point. This equation is crucial for understanding the behavior of functions in multivariable contexts, as it provides insight into instantaneous rates of change and local linear approximations of complex shapes.
Tangent Vector: A tangent vector is a vector that represents the direction and rate of change of a curve at a specific point. It captures how the curve is oriented in space and is essential for understanding the motion along the curve, connecting it to concepts like arc length and curvature, as well as derivatives and integrals of vector-valued functions.
Unit Tangent Vector: The unit tangent vector is a vector that points in the direction of the tangent to a curve at a given point and has a magnitude of one. It is derived from the derivative of a vector-valued function, representing how the curve is changing at that point. This vector provides important information about the direction of motion along the curve and is essential in understanding the geometric and physical properties of vector-valued functions.
Vector-valued functions: Vector-valued functions are mathematical functions that output vectors instead of single numerical values. These functions are crucial for describing curves and surfaces in higher-dimensional spaces, enabling the representation of motion, geometry, and various physical phenomena. By utilizing components in the form of vectors, we can analyze derivatives and integrals of these functions, as well as compute surface areas for parametric surfaces.
Work done by a force: Work done by a force is defined as the transfer of energy that occurs when a force acts on an object and causes it to move in the direction of the force. This concept is mathematically represented as the integral of the force along a path, capturing both the magnitude and direction of the force applied over a distance. The work done can vary based on the path taken and the nature of the force involved, whether it be constant or variable.
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