Mathematical Tools for Physics
8—Multivariable Calculus 213 Sometimes you see the chain rule written in a slightly different form. You can change coordinates f ...
8—Multivariable Calculus 214 8.4 Geometric Interpretation For one variable, the picture of the differential is simple. Start wit ...
8—Multivariable Calculus 215 For the case of three independent variables, I’ll leave the sketch to you. Examples The temperature ...
8—Multivariable Calculus 216 8.5 Gradient The equation ( 5 ) for the differential has another geometric interpretation. For a fu ...
8—Multivariable Calculus 217 For the examplef(x,y) =x^2 + 4y^2 ,G~= 2xˆx+ 8yˆy. At each point in thex-yplane it provides a vecto ...
8—Multivariable Calculus 218 is E~(x,y,z) =kq r^2 ˆr whereˆris the unit vector pointing away from the origin andris the distance ...
8—Multivariable Calculus 219 The symbol∇is commonly used for the gradient operator. This vector operator will appear in several ...
8—Multivariable Calculus 220 What is the total kinetic energy because of this oscillation? It is ∫ udA. To evaluate it, use pola ...
8—Multivariable Calculus 221 The surfaces that have constant values of these coordinates are planes in rectangular coordinates; ...
8—Multivariable Calculus 222 An Area Find the area in thex-yplane between the curvesy=x^2 /aandy=x. (A) ∫a 0 dx ∫x x^2 /a dy 1 a ...
8—Multivariable Calculus 223 A Moment of Inertia The moment of inertia about an axis is ∫ r⊥^2 dm. What is the moment of inertia ...
8—Multivariable Calculus 224 The element of area isR^2 sinθ dθ dφ, so the total charge is ∫ σ dA, Q= ∫π 0 sinθ dθ R^2 ∫ 2 π 0 dφ ...
8—Multivariable Calculus 225 increasing coordinateφ. Finallyθˆis perpendicular to the coneθ=constant and again, points toward in ...
8—Multivariable Calculus 226 nucleus are central to the subject of nuclear magnetic resonance (NMR), and that has its applicatio ...
8—Multivariable Calculus 227 8.11 Maxima, Minima, Saddles With one variable you can look for a maximum or a minimum by taking a ...
8—Multivariable Calculus 228 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 229 8.12 Lagrange Multipliers This is an incredibly clever method to handle problems of maxima and mini ...
8—Multivariable Calculus 230 To implement this picture so that you can compute with it, look at the gradient off and the gradien ...
8—Multivariable Calculus 231 The second example said that you have several different allowed energies, typical of what happens i ...
8—Multivariable Calculus 232 of several variables subject to constraints on N and on E. Now all you have to do is to figure out ...
«
7
8
9
10
11
12
13
14
15
16
»
Free download pdf