Electricity and Magnetism

Mathematical Tools and Fields

01Series Approximations: Keeping One or Two Terms 02Integrals as Sums: Along a Line and a Curved Path 03Line Integrals: Work as the Integral of F · ds 04Closed Line Integrals: Work Around a Loop 05Double Integrals: Area, Charge and Volume 06Closed Surface Integrals: The Sphere and Its Charge 07Electric Flux Through Surfaces and Volume Integrals Problem set0/10 Problem set 20/10 MIT problem set0/4

01Field Theory Versus Action at a Distance 02Scalar Fields: Contour, Color and Relief Maps 03Vector Fields, Sources, Sinks and Divergence 04Curl: Measuring the Circulation of a Field 05Gravitational, Electric and Magnetic Fields 06Drawing Vector Fields: Arrows, Lines and Grass Seeds Problem set0/10 Problem set 20/10 MIT problem set0/1

Integrals as Sums: Along a Line and a Curved Path

How does reading an integral sign as a sum of tiny pieces carry you from the x-axis to any curved path?

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Your summary note

1. The integral as a limit of a sum: $L = \int_a^b dx$
  Write the length of a segment from $x=a$ to $x=b$ as $\sum_i \Delta x_i$ , pass to the limit $\Delta x \rightarrow dx$ , and get $L = b - a$ in meters.
2. Total charge from a linear charge density $\lambda$
  Define $\lambda$ in C/m, compute $Q = \lambda(b-a)$ for uniform charge, then integrate $\lambda(x) = kx^2$ to get $Q = \frac{k(b^3-a^3)}{3}$ , and write the limit-of-sum definition $\sum_i \lambda(x_i)\,\Delta x_i$ .
3. The line integral along a curved path
  Sketch a curve from $a$ to $b$ with the displacement vector $\mathbf{ds}$ tangent to it, state what its magnitude and direction are, and write $L$ as the sum of the $|\mathbf{ds}|$ .
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