Mathematical Tools for Physics - Department of Physics - University

9—Vector Calculus 1 214 whereαis the angle between the direction of the fluid velocity and the normal to the area. C~ D~ θ C~×D~ ...

9—Vector Calculus 1 215 y x φ ˆnk b Now to implement the calculation of the flow rate: Divide the area intoNpieces of length∆`ka ...

9—Vector Calculus 1 216 through this piece of the surface is ~vk.∆A~k=v 0 yk b xˆ.a b 2 ∆θkˆnk The value ofykat the angleθkis yk ...

9—Vector Calculus 1 217 How can you do this for fluid flow? If you inject a small amount of dye into the fluid at some point it ...

9—Vector Calculus 1 218 viscosity, but I’ll put it aside for now save for one observation: how much information is needed to des ...

9—Vector Calculus 1 219 general expression for the rate of change of volume in a surface being carried with the fluid. It’s also ...

9—Vector Calculus 1 220 of terms to integrate, at least they’re all easy. Take the first of them: ∫y 0 +∆y y 0 dy ∫z 0 +∆z z 0 d ...

9—Vector Calculus 1 221 The symbol∇will take other forms in other coordinate systems. Now that you’ve waded through this rather ...

9—Vector Calculus 1 222 The corresponding expression in spherical coordinates is found in exactly the same way, prob- lem9.4. di ...

9—Vector Calculus 1 223 ∮ dA~×~v= ∮ ~ωRdA−~rωdAcosθ =~ωR 4 πR^2 −ω ∫π 0 R^2 sinθdθ ∫ 2 π 0 dφzRˆ cosθcosθ =~ωR 4 πR^2 −ωzˆ 2 πR^ ...

9—Vector Calculus 1 224 9.5 The Gradient The gradient is the closest thing to an ordinary derivative here, taking a scalar-value ...

9—Vector Calculus 1 225 This agrees with equation (9.15). Similarly you can use the results of problem8.15to find the derivative ...

9—Vector Calculus 1 226 density,dm/dV of the matter that is generating the gravitational field. Gis Newton’s gravitational const ...

9—Vector Calculus 1 227 Solve forCand you have C=− 4 3 πGρ 0 R^3 =− 4 3 πG 3 M 4 πR^3 R^3 =−GM Put this all together and express ...

9—Vector Calculus 1 228 This is a scalar equation instead of a vector equation, so it will often be easier to handle. Apply it t ...

9—Vector Calculus 1 229 V dV/dr d^2 V/dr^2 The second derivative on the left side of Eq. (9.44) has a double spike that does not ...

9—Vector Calculus 1 230 Magnetic Boundary Conditions The equations for (time independent) magnetic fields are ∇×B~=μ 0 J~ and ∇. ...

9—Vector Calculus 1 231 The Kronecker delta is either one or zero depending on whetheri=jori 6 =j, and this equation sums up the ...

9—Vector Calculus 1 232 If the indices are a cyclic permutation of 123, (231 or 312), the alternating symbol is 1. If the indice ...

9—Vector Calculus 1 233 9.11 More Complicated Potentials The gravitational field from a point mass is~g=−Gmˆr/r^2 , so the poten ...