Mathematical Tools for Physics
9—Vector Calculus 1 273 positions of the charges, and the non-primed ones are the position of the point where you are evaluating ...
9—Vector Calculus 1 274 Problems 9.1 Use the same geometry as that following Eq. ( 3 ), and take the velocity function to be~v=x ...
9—Vector Calculus 1 275 9.8 What is the area of the spherical cap on the surface of a sphere of radiusR: 0 ≤θ≤θ 0? (b) Does the ...
9—Vector Calculus 1 276 9.15 If you have a very large (assume it’s infinite) slab of mass of thicknessdthe gravitational field w ...
9—Vector Calculus 1 277 A uniformly charged ball of radiusRhas charge densityρ 0 forr < R, Q= 4πρ 0 R^3 / 3. What is the elec ...
9—Vector Calculus 1 278 9.25 A fluid of possibly non-uniform mass density is in equilibrium in a possibly non-uniform gravitatio ...
9—Vector Calculus 1 279 9.28 In the preceding problem, what is the total energy in the gravitational field, ∫ udV? How does this ...
9—Vector Calculus 1 280 and graph the potential functionV(r)for this limiting case. This violates Eq. ( 42 ). Why? (c) Compute t ...
9—Vector Calculus 1 281 9.41 Compute the divergence and the curl of yˆx−xˆy x^2 +y^2 , and of yxˆ−xyˆ (x^2 +y^2 )^2 9.42 Transla ...
9—Vector Calculus 1 282 9.47 Fill in the missing steps in deriving Eq. ( 45 ). 9.48 Analyze the behavior of Eq. ( 45 ). (a) Ifz= ...
Partial Differential Equations If the subject of ordinary differential equations is large, this is enormous. I am going to exami ...
10—Partial Differential Equations 284 For a slab of areaA, thickness∆x, and mass density ρ, let the coordinates of the two sides ...
10—Partial Differential Equations 285 whereH~ is the heat flow vector, the power per area in the direction of the energy transpo ...
10—Partial Differential Equations 286 Denote the constant byκ/Cρ=Dand divide by the productfg. 1 f df dt =D 1 g d^2 g dx^2 (7) T ...
10—Partial Differential Equations 287 It is only the combined product that forms a solution to the original partial differential ...
10—Partial Differential Equations 288 Multiply bysin ( mπx/L ) and integrate over the domain to isolate the single term,n=m. ∫L ...
10—Partial Differential Equations 289 The differential equation for the temperature is still Eq. ( 3 ), and I’ll assume that the ...
10—Partial Differential Equations 290 The further condition is that atx= 0the temperature isT 1 e−iωt, so that tells you thatB=T ...
10—Partial Differential Equations 291 O 0 b y 0 T 0 a 0 x Look at this problem from several different angles, tear it apart, loo ...
10—Partial Differential Equations 292 The accompanying equation forgis now d^2 g(y) dy^2 = +k^2 g =⇒ g(y) =Csinhky+Dcoshky (Or e ...
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