Mathematical Tools for Physics
9—Vector Calculus 1 253 And this is the same result that I got for the flat surface calculation. I set it up so that the two res ...
9—Vector Calculus 1 254 How can I get some simpler picture? Do it in the same spirit that you introduce the derivative: Concentr ...
9—Vector Calculus 1 255 tells me about the rotation of the fluid in the immediate neighborhood of that point. If I place a tiny ...
9—Vector Calculus 1 256 the motion isn’t likely to be perpendicular to the surface. It’s only thecomponentof the velocity normal ...
9—Vector Calculus 1 257 The Divergence as Derivatives This is still a long way from something that you can easily compute. I’ll ...
9—Vector Calculus 1 258 they’re all easy. Take the first of them: ∫y 0 +∆y y 0 dy ∫z 0 +∆z z 0 dz [ vx( 0 )+ (∆x) ∂vx ∂x ( 0 ) + ...
9—Vector Calculus 1 259 In the limit that the all the∆x,∆y, and∆zshrink to zero the terms with a second derivative vanish, as do ...
9—Vector Calculus 1 260 steps still more. Do this for the other sides, add, and you get the result. It all looks very simple whe ...
9—Vector Calculus 1 261 These are the three commonly occurring coordinates system, though the same simplified method will work i ...
9—Vector Calculus 1 262 Choose a spherical coordinate system with thez-axis along~ω. dA~=ˆndA=r dA,ˆ and ~ω.dA~=ω dAcosθ ∮ dA~×~ ...
9—Vector Calculus 1 263 When you subtract the second from the first and divide by the volume,∆x∆y∆z, what is left is (in the lim ...
9—Vector Calculus 1 264 cylindrical: ∇=ˆr ∂ ∂r +ˆθ 1 r ∂ ∂θ +ˆz ∂ ∂z (24) spherical: ∇=ˆr ∂ ∂r +ˆθ 1 r ∂ ∂θ +φˆ 1 rsinθ ∂ ∂φ In ...
9—Vector Calculus 1 265 This agrees with equation ( 15 ). Similarly you can use the results of problem8.15to find the derivative ...
9—Vector Calculus 1 266 9.8 Applications to Gravity The basic equations to describe the gravitational field in Newton’s theory a ...
9—Vector Calculus 1 267 Outside the surfacer=R, the mass density is zero, so this is 1 r^2 d ( r^2 gr ) dr = 0, implying r^2 gr= ...
9—Vector Calculus 1 268 9.9 Gravitational Potential The gravitational potential is that functionV for which ~g=−∇V (36) That suc ...
9—Vector Calculus 1 269 Inside: 1 r^2 d dr ( r^2 dV dr ) = 4πGρ 0 so r^2 dV dr = 4πGρ 0 r^3 3 +C′ Continue, dividing byr^2 and i ...
9—Vector Calculus 1 270 The second derivative on the left side of Eq. ( 39 ) has a double spike that does not appear on the righ ...
9—Vector Calculus 1 271 Did I say that the use of potentials is supposed to simplify the problems? Yes, but only the harder prob ...
9—Vector Calculus 1 272 9.11 More Complicated Potentials The gravitational field from a point mass is~g=−Gmr/rˆ^2 , so the poten ...
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