Mathematical Tools for Physics - Department of Physics - University

8—Multivariable Calculus 194 Notice that the units work out right too. In spherical coordinates the procedure is identical. All ...

8—Multivariable Calculus 195 These two equations determine the parametersαandβ. α ∫L 0 dxsin^2 πx L = ∫L 0 dxf(x) sin πx L β ∫L ...

8—Multivariable Calculus 196 Write this in a more compact notation in order to emphasize the important parts. f(~r+d~r)−f(~r) =∇ ...

8—Multivariable Calculus 197 I’ll describe this procedure for two variables; it’s the same for more. The problem stated is that ...

8—Multivariable Calculus 198 The equations to solve are now ∇(f−λφ) = 0, and φ= 0, which become 4 y−λ 2 x a^2 = 0, 4 x−λ 2 y b^2 ...

8—Multivariable Calculus 199 This, I can differentiate. Maximizing (8.36) is the same as maximizing its logarithm, and that’s ea ...

8—Multivariable Calculus 200 8.13 Solid Angle The extension of the concept of angle to three dimensions is called “solid angle.” ...

8—Multivariable Calculus 201 whereRis the rate of entry andσis the area of the hole. If I can measure the rate of absorptionR an ...

8—Multivariable Calculus 202 integral over all the azimuthal anglesφaround the ring just gives a factor 2 π. The element of soli ...

8—Multivariable Calculus 203 middle one has equal angles of incidence and reflection. The dashed lines are from the center of th ...

8—Multivariable Calculus 204 To complete this I should finish withdσ/dΩ. The interesting part of the problem is near the vertica ...

8—Multivariable Calculus 205 Exercises 1 For the functionsf(x,y) =Axy^2 sin(xy), x(t) =Ct^3 , y(t) =Dt^2 , computedf/dttwo ways. ...

8—Multivariable Calculus 206 Problems 8.1 Letr= √ x^2 +y^2 ,x=Asinωt,y=Bcosωt. Use the chain rule to compute the derivative with ...

8—Multivariable Calculus 207 simpler. Don’t stop with just the field at fixedras I suggested you begin. The field fills space, s ...

8—Multivariable Calculus 208 8.16 Compute the volume of a sphere using spherical coordinates. Also do it using rectangular coor- ...

8—Multivariable Calculus 209 8.26 In problem2.26you computed the electric potential at large distances from a set of three charg ...

8—Multivariable Calculus 210 the relation between the impact parameterband the scattering angleθ. Then compute the differential ...

8—Multivariable Calculus 211 constraints is on the edge of this allowed region, and that it is at the corner of intersection of ...

8—Multivariable Calculus 212 Compute the total kinetic and the total potential energy for this oscillation, a function of time. ...

Vector Calculus 1 . The first rule in understanding vector calculus isdraw lots of pictures. This subject can become rather abst ...