MODULE 3

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• Lesson 3 - Vector Calculus

  • Vector Functions
  • Limits, Derivative, and Integration of Vector Function
  • Motion on a Curve and Curvature on a Smooth Curve
  • Tangent, Normal and Binormal Unit Vector
  • Differential Length, Area, and Volume
  • Line, Surface, and Volume Integrals
  • Del Operator and Gradient of a Scalar
  • Divergence of a Vector and Divergence Theorem
  • Curl of a Vector and Stokes' Theorem
  • Laplacian of a Scalar
  • Classification of Vector Fields
• Recall that a curve C in the xy-plane is simply a set of ordered pairs (x, y).
• Parametric curve

  A curve where the x- and y-coordinates of a point on the curve are defined by a pair of functions x = f(t), y = g(t) that are continuous on some interval a ≤ t ≤ b
• Space curve

  A parametric curve in 3-space where x = f(t), y = g(t), z = h(t) are continuous on an interval a ≤ t ≤ b
• Vector-valued function

  A vector r whose components are functions of a parameter t, r(t) = <f(t), g(t), h(t)> or r(t) = f(t)i + g(t)j + h(t)k
• If lim t→a f(t), lim t→a g(t), and lim t→a h(t) exist, then lim t→a r(t) = <lim t→a f(t), lim t→a g(t), lim t→a h(t)>
• Continuous vector function

  r(t) is continuous at t = a if i) r(a) is defined, ii) lim t→a r(t) exists, and iii) lim t→a r(t) = r(a)
• The derivative of a vector function r is r'(t) = lim (t→a) (1/Δt)[r(t+Δt) - r(t)] for all t where the limit exists
• If r(t) = <f(t), g(t), h(t)>, where f, g, and h are differentiable, then r'(t) = <f'(t), g'(t), h'(t)>
• The integral of a vector function r is ∫a^b r(t) dt = ∫a^b f(t) dt i + ∫a^b g(t) dt j + ∫a^b h(t) dt k
• The length of a smooth space curve traced by r(t) = <f(t), g(t), h(t)> is s = ∫a^b |r'(t)| dt
• Motion on a Curve

  1. Position vector r(t) = f(t)i + g(t)j + h(t)k

  2. Velocity vector v(t) = r'(t) = f'(t)i + g'(t)j + h'(t)k

  3. Acceleration vector a(t) = r''(t) = f''(t)i + g''(t)j + h''(t)k
• Speed is related to arc length s by s'(t) = |v(t)| and s = ∫t0^t1 |v(t)| dt
• Unit tangent vector

  T(t) = r'(t)/|r'(t)|
• Curvature

  κ = |T'(t)| / |r'(t)| where T = dr/ds is the unit tangent vector
• The curvature of a circle of radius a is 1/a
• Curvature of a circle

  Reciprocal of the radius of the circle
• The curvature at a point on a circle indicates that a circle with a small radius curves more than one with a large radius
• Acceleration vector and its components

  1. Differentiating the equation for velocity

  2. Tangential component (change in magnitude of velocity)

  3. Normal component (change in direction of velocity)
• Tangential component of acceleration

  𝒂𝑻 = 𝒅𝒗/𝒅𝒕
• Normal component of acceleration

  𝒂𝑵 = 𝜿𝒗𝟐
• Curvature

  𝜿 = |𝒗 × 𝒂|/𝒗𝟑 = |𝒓′ 𝒕 × 𝒓′′ 𝒕 |/|𝒓′ 𝒕 |𝟑
• Binormal vector

  𝑩 𝒕 = 𝑻(𝒕) × 𝑵(𝒕)
• The three unit vectors (tangent, normal, and binormal) form a right-handed set of mutually orthogonal vectors called the moving trihedral
• The plane of unit tangent vector and unit normal vector is called the osculating plane, the plane of unit normal vector and binormal vector is said to be the normal plane, and the plane of unit tangent vector and binormal vector is the rectifying plane
• Finding unit tangent vector and unit normal vector

  From position function 𝒓(𝒕)
• Finding tangential and normal components of acceleration, unit tangent vector, unit normal vector, binormal vector, and curvature

  From position function 𝒓(𝒕)
• Finding unit tangent vector, unit normal vector, binormal vector, curvature, and equations of osculating, normal, and rectifying planes

  From position function 𝒓(𝒕)
• Differential displacement

  𝒅𝒍 = 𝒅𝒙 𝒂𝒙 + 𝒅𝒚 𝒂𝒚 + 𝒅𝒛 𝒂𝒛
• Differential normal surface area

  𝒅𝑺 = 𝒅𝒚 𝒅𝒛 𝒂𝒙, 𝒅𝑺 = 𝒅𝒙 𝒅𝒛 𝒂𝒚, 𝒅𝑺 = 𝒅𝒙 𝒅𝒚 𝒂𝒛
• Differential volume

  𝒅𝑽 = 𝒅𝒙 𝒅𝒚 𝒅𝒛
• Finding length, area, and volume of an object

  Using differential formulas in Cartesian, cylindrical, and spherical coordinate systems
• Line

  Path along a curve in space
• Line integral

  The integral of the tangential component of A along curve L
• Closed contour integral

  The circulation of A around L
• Surface integral

  The flux of A through S
• Net outward flux

  The surface integral over a closed surface (defining a volume)
• Volume integral

  The integral of the scalar ρV over the volume V
• Del operator (∇)

  The vector differential operator