Scalar and vector valued functions

Scalar and vector valued functions

1. Two Kinds of Functions in ℝ³

• Scalar-valued (scalar field / point function):

assigns a number to each point — e.g., temperature $T(x,y,z)$, pressure, potential $φ = x²+y²+z²$, density.

• Vector-valued (vector field):

assigns a vector to each point — e.g., velocity $V$, force $F$, electric field $E$, gradient $∇φ$.

A vector-valued function of one variable $r(t)= x(t)i + y(t)j + z(t)k$ traces a curve $C$ (parametric trajectory). A scalar field defines level surfaces $φ = c$ (isotherms, equipotentials); a vector field defines field lines (tangent to $F$ at each point).

2. Notation and Representation

• $i, j, k$ = unit vectors along $x,y,z$.
• $r = x i + y j + z k$ = position vector of point $P(x,y,z)$ with $|r|=√(x²+y²+z²)$.
• $F = <P, Q, R>$ component form; $P,Q,R$ are scalar fields.

3. Operations Preview (roadmap to gradient/divergence/curl)

• Gradient $∇φ$ turns scalar → vector (direction of steepest ascent).
• Divergence $∇·F$ turns vector → scalar (source density).
• Curl $∇×F$ turns vector → vector (rotation density).
  All require $P,Q,R, φ$ have first partials.

4. Worked Examples

(a) φ(x,y,z)= x² + y² + z² : scalar field; level surfaces φ=c are spheres radius √c ✓

(b) F(x,y,z)= x i + y j + z k = r : vector field radially outward; field lines are rays from origin ✓

(c) Velocity field of rotating disc: F= -y i + x j (2D): vectors tangent to circles, magnitude = r ✓

(d) r(t)= a cos t i + a sin t j + b t k : vector-valued of t = helix radius a, pitch 2πb ✓

(e) Temperature T(x,y,z)=100 e^{-(x²+y²+z²)} : scalar field; gradient points toward origin (coldest? Actually T decreases outward, so -∇T points outward) ✓

5. Domain and Continuity

$φ$ and $F$ are defined on a domain $D⊂ℝ³$; continuity of each component ⇔ continuity of the field. Level surfaces $φ=c$ and field lines are geometric objects used in integration (line, surface, volume integrals next).

6. Exam Points

• Define and distinguish scalar vs vector point functions with examples — 3 marks, very frequent.
• Write $r = x i + y j + z k$ and $F = P i+Q j+R k$ — 3 marks.
• Identify level surfaces vs field lines for given $φ,F$ — 3-5 marks, frequent.
• State that $r(t)$ is vector-valued of one variable tracing a curve — 3 marks.