Vector Calculus 1
Monday, 10 August 2026 02:07
Scalars, vectors, fields and functions
• A scalar function (of one variable) g(x) or g(t) is a formula that takes a scalar and returns a scalar → Scalar
is a quantity defined by its magnitude only, e.g. Time (10s), Mass (5kg) → Vector is magnitude plus direction
• It might be used to describe the spatial variation of temperature T(x) along a 1-D bar heated at one end, or
the time variation of the DC current i(t) across a certain component in an electrical circuit
• g: ℝ → ℝ (input: real number, output: real number) → ‘:’ = ‘in terms of’ or ‘mapping to’
• A vector function (of one variable) V(x) or V(t) takes a scalar and returns a vector: V(t) = V1(t)i + V2(t)j +
V3(t)k → or alternatively (V₁(t), V₂(t), V₃(t))
• Such functions might be used to describe the motion of a particle whose position vector is r(t) at time t, or
the external forces F(x) acting at distance x along a 1D strut
• V: ℝ → ℝ³ (input: real number, output: three real numbers (vector))
• A scalar field φ is a scalar quantity defined over a region of space
• It takes a vector (of positions) and returns a scalar: φ = g(x,y,z) = g(r) (or g(x,y)) → e.g. T(x,y) = 100 − 2x³ −
2y² → input is a vector, output is a scalar
• Such functions might be used to describe the variation of temperature T(x,y,z) in the room using Cartesian
coordinates, or the variation of density/charge density ρ(x,y,z) inside a solid object
• φ: ℝ³ → ℝ (input: three real numbers (coordinates), output: one real number)
• A vector field V(x,y,z) is a vector-valued quantity defined over a region of space
• It is defined by a field that takes a vector (of positions) and returns a vector: V(r) = V₁(x,y,z)i + V₂(x,y,z)j
+ V₃(x,y,z)k (in 3D)
• Such functions might be used to describe the spatial variation of fluid velocity v(x,y,z) in a steady flow, or
current I(x,y,z) flowing in a conductor
• V: ℝ³ → ℝ³ (in 3D)
Vector functions (V: ℝ → ℝ³)
• Differentiation and integration of vector functions are easy!
• Simply differentiate/integrate the components separately: d/dt V(t) = d/dt V₁(t)i + d/dt V ₂(t)j + d/dt V ₃(t)k
• The following rules of differentiation are obtained by applying the corresponding rules to the separate
components
• Here U = U(t), V = V(t), ′ = d/dt , and c is a constant:
◦ (U + V)′ = U′ + V′
◦ (cU)′ = cU′
◦ Product rules for vector functions:
▪ (U·V)′ = U′·V + U·V′
▪ (U×V)′ = U′×V + U×V′
• One important type of vector function is the intrinsic definition of a curve r(t) in 3 dimensions: r(t) = (x(t),
y(t), z(t)) = x(t)i + y(t)j + z(t)k
• e.g. Straight line r = a + tb ; circle r = (a cos(t), a sin(t), 0) in the (x,y)-plane
• The tangent vector to the curve r(t): r′(t) = (x′(t), y′(t), z′(t))
• There are many ways of choosing the parametrisation variable t (replacing t by t² doesn’t change the curve,
nor the direction of the tangent vector)
• An important one is the so-called distance, or arclength s along the curve → refer to Line integrals (week 4)
Scalar fields (φ: ℝ³ → ℝ , or ℝ² → ℝ)
• How do we visualise scalar fields φ = g(r)?
Monday, 10 August 2026 02:07
Scalars, vectors, fields and functions
• A scalar function (of one variable) g(x) or g(t) is a formula that takes a scalar and returns a scalar → Scalar
is a quantity defined by its magnitude only, e.g. Time (10s), Mass (5kg) → Vector is magnitude plus direction
• It might be used to describe the spatial variation of temperature T(x) along a 1-D bar heated at one end, or
the time variation of the DC current i(t) across a certain component in an electrical circuit
• g: ℝ → ℝ (input: real number, output: real number) → ‘:’ = ‘in terms of’ or ‘mapping to’
• A vector function (of one variable) V(x) or V(t) takes a scalar and returns a vector: V(t) = V1(t)i + V2(t)j +
V3(t)k → or alternatively (V₁(t), V₂(t), V₃(t))
• Such functions might be used to describe the motion of a particle whose position vector is r(t) at time t, or
the external forces F(x) acting at distance x along a 1D strut
• V: ℝ → ℝ³ (input: real number, output: three real numbers (vector))
• A scalar field φ is a scalar quantity defined over a region of space
• It takes a vector (of positions) and returns a scalar: φ = g(x,y,z) = g(r) (or g(x,y)) → e.g. T(x,y) = 100 − 2x³ −
2y² → input is a vector, output is a scalar
• Such functions might be used to describe the variation of temperature T(x,y,z) in the room using Cartesian
coordinates, or the variation of density/charge density ρ(x,y,z) inside a solid object
• φ: ℝ³ → ℝ (input: three real numbers (coordinates), output: one real number)
• A vector field V(x,y,z) is a vector-valued quantity defined over a region of space
• It is defined by a field that takes a vector (of positions) and returns a vector: V(r) = V₁(x,y,z)i + V₂(x,y,z)j
+ V₃(x,y,z)k (in 3D)
• Such functions might be used to describe the spatial variation of fluid velocity v(x,y,z) in a steady flow, or
current I(x,y,z) flowing in a conductor
• V: ℝ³ → ℝ³ (in 3D)
Vector functions (V: ℝ → ℝ³)
• Differentiation and integration of vector functions are easy!
• Simply differentiate/integrate the components separately: d/dt V(t) = d/dt V₁(t)i + d/dt V ₂(t)j + d/dt V ₃(t)k
• The following rules of differentiation are obtained by applying the corresponding rules to the separate
components
• Here U = U(t), V = V(t), ′ = d/dt , and c is a constant:
◦ (U + V)′ = U′ + V′
◦ (cU)′ = cU′
◦ Product rules for vector functions:
▪ (U·V)′ = U′·V + U·V′
▪ (U×V)′ = U′×V + U×V′
• One important type of vector function is the intrinsic definition of a curve r(t) in 3 dimensions: r(t) = (x(t),
y(t), z(t)) = x(t)i + y(t)j + z(t)k
• e.g. Straight line r = a + tb ; circle r = (a cos(t), a sin(t), 0) in the (x,y)-plane
• The tangent vector to the curve r(t): r′(t) = (x′(t), y′(t), z′(t))
• There are many ways of choosing the parametrisation variable t (replacing t by t² doesn’t change the curve,
nor the direction of the tangent vector)
• An important one is the so-called distance, or arclength s along the curve → refer to Line integrals (week 4)
Scalar fields (φ: ℝ³ → ℝ , or ℝ² → ℝ)
• How do we visualise scalar fields φ = g(r)?
