Key Concepts of Vector-Valued Functions

Why This Matters

Vector-valued functions are your bridge from single-variable calculus to the multidariable world—and they're essential for describing anything that moves through space. When you're tested on this material, you're not just being asked to compute derivatives and integrals component-by-component. You're being tested on whether you understand how position, velocity, and acceleration relate to each other, how parametric representations free us from the limitations of $y = f(x)$, and how geometric properties like arc length and curvature emerge from calculus operations.

These concepts connect directly to physics applications you'll see on exams: projectile motion, circular motion, and work done by forces. The key insight is that vector-valued functions let us treat multidimensional problems using the same differentiation and integration techniques you've already mastered—just applied component-wise. Don't just memorize formulas—know why the derivative gives velocity, why we integrate speed (not velocity) for arc length, and how tangent and normal vectors describe a curve's geometry.

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Foundations: Defining and Representing Curves

Before you can analyze motion, you need to understand how vector-valued functions encode curves in space. The fundamental idea is that a single parameter (usually time) controls the position vector's components simultaneously.

Definition of Vector-Valued Functions

• A vector-valued function maps real numbers to vectors—written as $\mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle$, where each component is a scalar function
• The domain is typically an interval of real numbers, while the output lives in $\mathbb{R}^2$ or $\mathbb{R}^3$—this is what distinguishes vector-valued functions from scalar functions
• The output traces a curve in space as $t$ varies—think of $t$ as time and $\mathbf{r}(t)$ as the position of a moving particle

Parametric Equations and Curves

• Each component function is a parametric equation—$x(t)$, $y(t)$, and $z(t)$ work together to locate points on the curve
• Parametric form handles curves impossible to write as $y = f(x)$—circles, spirals, and self-intersecting curves all become straightforward
• The parameter doesn't have to be time—it could be angle, arc length, or any convenient variable that traces the curve

Limits and Continuity of Vector-Valued Functions

• Evaluate limits component-by-component—$\lim_{t \to a} \mathbf{r}(t) = \langle \lim_{t \to a} x(t), \lim_{t \to a} y(t), \lim_{t \to a} z(t) \rangle$
• Continuity requires all components to be continuous—a single discontinuous component makes the entire vector function discontinuous
• Discontinuities create breaks or jumps in the curve—essential for understanding where parametric curves are well-behaved

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Differentiation: From Position to Motion

Differentiation of vector-valued functions unlocks the physics of motion. The derivative tells you how the position vector changes, which directly gives velocity and direction of travel.

Derivatives of Vector-Valued Functions

• Differentiate component-by-component—if $\mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle$, then $\mathbf{r}'(t) = \langle x'(t), y'(t), z'(t) \rangle$
• The derivative represents instantaneous rate of change of position—this is the mathematical definition of velocity
• Product rules extend naturally—for $\mathbf{r}(t) \cdot \mathbf{s}(t)$ and $\mathbf{r}(t) \times \mathbf{s}(t)$, apply the product rule but preserve vector operation order for cross products

Velocity and Acceleration Vectors

• Velocity is the first derivative: $\mathbf{v}(t) = \mathbf{r}'(t)$—it gives both speed (magnitude) and direction of motion
• Acceleration is the second derivative: $\mathbf{a}(t) = \mathbf{v}'(t) = \mathbf{r}''(t)$—it describes how velocity changes over time
• Speed is the magnitude of velocity: $|\mathbf{v}(t)| = \sqrt{[x'(t)]^2 + [y'(t)]^2 + [z'(t)]^2}$—a scalar, not a vector

Tangent Vectors and Normal Vectors

• The tangent vector $\mathbf{T}(t) = \frac{\mathbf{r}'(t)}{|\mathbf{r}'(t)|}$—a unit vector pointing in the direction of motion
• The principal normal vector $\mathbf{N}(t) = \frac{\mathbf{T}'(t)}{|\mathbf{T}'(t)|}$—perpendicular to $\mathbf{T}$, pointing toward the center of curvature
• Together, $\mathbf{T}$ and $\mathbf{N}$ form the osculating plane—the plane that best approximates the curve at that point

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Integration and Arc Length: Accumulation Along Curves

Integration reverses differentiation and enables us to measure curves. The key insight is that integrating velocity gives displacement, while integrating speed gives distance traveled.

Integrals of Vector-Valued Functions

• Integrate component-by-component—$\int \mathbf{r}(t)\, dt = \langle \int x(t)\, dt, \int y(t)\, dt, \int z(t)\, dt \rangle + \mathbf{C}$
• Definite integrals give displacement vectors—$\int_a^b \mathbf{v}(t)\, dt = \mathbf{r}(b) - \mathbf{r}(a)$, the net change in position
• Don't confuse displacement with distance—displacement is a vector (can be zero for closed paths); distance is always positive

Arc Length and Curvature

• Arc length formula: $L = \int_a^b |\mathbf{r}'(t)|\, dt = \int_a^b \sqrt{[x'(t)]^2 + [y'(t)]^2 + [z'(t)]^2}\, dt$
• Curvature measures bending: $\kappa = \frac{|\mathbf{T}'(t)|}{|\mathbf{r}'(t)|} = \frac{|\mathbf{r}'(t) \times \mathbf{r}''(t)|}{|\mathbf{r}'(t)|^3}$—higher $\kappa$ means sharper turns
• Arc length parametrization uses $s$ (distance along curve) as the parameter—this makes $|\mathbf{r}'(s)| = 1$ always

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Vector Fields: Functions of Position

Vector fields extend the idea of vector-valued functions by assigning vectors to points in space rather than to parameter values. Instead of tracing a single curve, a vector field describes a quantity (like force or fluid velocity) throughout a region.

Vector Fields

• A vector field assigns a vector to each point: $\mathbf{F}(x, y, z) = \langle P(x,y,z), Q(x,y,z), R(x,y,z) \rangle$
• Visualize as arrows throughout space—the arrow at each point shows direction and magnitude of the field there
• Common examples include gravitational fields, electric fields, and fluid velocity fields—any quantity with both magnitude and direction at every location

Divergence and Curl of Vector Fields

• Divergence measures "spreading out": $\text{div } \mathbf{F} = \nabla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}$—positive means source, negative means sink
• Curl measures "rotation": $\text{curl } \mathbf{F} = \nabla \times \mathbf{F}$—nonzero curl indicates the field has rotational tendency
• Both are essential for physics applications—divergence appears in continuity equations; curl appears in electromagnetic theory and fluid vorticity

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Quick Reference Table

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Self-Check Questions

1. If a particle returns to its starting point after traveling along a curve, what can you say about its displacement vector versus its total distance traveled?

2. Which two quantities—velocity, speed, tangent vector, or acceleration—are always parallel to each other? Which is always a scalar?

3. A curve has curvature $\kappa = 0$ everywhere. What does this tell you about the shape of the curve?

4. Compare and contrast: How does differentiating a vector-valued function differ from differentiating a scalar function? What stays the same?

5. If a vector field has positive divergence at a point but zero curl, describe what's happening physically at that point. Give an example of such a field.