By Roel Snieder

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The area enclosed by the contour is given by rdθdr. 3) that the ϕ-component of ∇ × v is given by: 1 {vθ (r + dr, θ)(r + dr)dθ − vr (r, θ + dθ)dr − vθ (r, θ)rdθ + vr (r, θ)dr} . 14) In this expression vr and vθ denote the components of v in the radial direction and ˆ respectively. 6. 14). This result can be simplified by Taylor expanding the components of v in dr and dθ and linearizing the resulting expression in the infinitesimal increments dr and dθ. Problem b: Do this and show that the final result does not depend on dr and dθ and is given by: 1 ∂vr 1 ∂ (rvθ ) − .

We know that we live in a world of three spatial dimensions, but it is interesting to investigate if the orbit of the earth would also be stable in a world with a different number of spatial dimensions. 67 × 10−8 cm3 g−1 s−2 . 12). 12). This is due to the fact that two electric charges of equal sign repel each other, while two masses of equal sign (mass being positive) attract each other. 17) would be positive, masses would repel each other and structures such as planets, the solar system and stellar systems would not exist.

THE THEOREM OF STOKES bounded by the contour over which the line integration is carried out. This is a very useful result because often the surface integration can be simplified by choosing the surface carefully. 2 Stokes’ theorem from the theorem of Gauss Stokes’ law is concerned with surface integrations. Since the curl is intrinsically a threedimensional vector, Stokes’s law is inherently related to three space dimensions. However, if we consider a vector field that depends only on the coordinates x and y (v = v(x, y)) and that has a vanishing component in the z-direction (vz = 0), then ∇ × v points along the z-axis.

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A Guided Tour of Mathematical Physics by Roel Snieder
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