16.1 Vector Fields

Vector Fields: Functions with n inputs and an n-dimentional vector output

• $F(x,y) = \langle a, b\rangle = \lang P(x,y), Q(x,y) \rang$ (P and Q are called “scalar fields”)

Vector Field Applications

• Velocity Field: When V(…) = the velocity vector at that point
• Gravitational Field: $F(\vec x)= -\frac{mMG}{|\vec x |^3}\vec x$, each point is a force.
• Electric Force Field: $E(\vec x) = \frac{1}{q}F(\vec x)=\frac{\epsilon Q}{|\vec x|^3}\vec x$

Gradient Vector Field: $\nabla f(\vec v)= \lang f_x(\vec v), f_y(\vec v), f_z(\vec v) \rang$

16.2 Line Integrals (Parametric)

Line Integral wrt Arc Len: Integrate f over a curve C instead of an interval [a, b]. Smooth Curve: Curve r such that r’ is continuous and r’(t) ≠ 0

• Let $x=g(t),y=h(t),z=f(x,y)$ be a 3D line plane
• Riemann Form: $\int_Cf(x,y) \, ds = \lim_{n\to \infty} \sum_{i=1}^n f(x_i^, y_i^) \Delta s_i$ ($\Delta s_i$ is the arc length change triggered by a small change in t)
• $\int_Cf \, ds = \int_{t=a}^{t=b}f(x(t), y(t)) \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} \, dt$

Line Integral Applications

• Center of Mass: If ρ is the density
  • Mass: $m=\int_c \rho(x,y) \, ds$
  • CoM at: $\bar x=\frac{1}{m}\int_c x \, \rho(x,y) \, ds, \, \bar y = \dots$

Line Integral wrt x and y:

• $\int_C f \, dx = \int_a^bf(x, y) x'(t) \, dt$ (Remember to add the x’ term!)
• Commonly written $\int_C P \, dx + Q \, dy$
• Changes for different orientations of the same curve

Line Integral of Vector Field