Exercises in Environmental Physics by Valerio Faraoni

By Valerio Faraoni

The research of environmental physics calls for realizing subject matters from many di?erent components of physics in addition to comprehension of phy- cal points of the realm round us. a number of first-class textbooks can be found overlaying such a lot points of environmental physics and of appli- tions of physics to the average atmosphere from a variety of issues of view. although, whereas instructing environmental physics to school scholars, I sorelymissedabookspeci?callydevotedtoexercisesfortheenvironm- tal technology pupil. hence, the incentive for this publication happened as in physics, in addition to in lots of different disciplines, passable wisdom of a topic can't be received with no perform. often scholars are usually not accustomed to many of the parts of physics which are required to explain either the surroundings and the human impression upon it. even as, scholars have to enhance talents within the manipulation of the tips and c- cepts realized in school. for that reason, this workout booklet is addressed to all degrees of college scholars in environmental sciences. end result of the wide variety of strength clients this e-book comprises either calculus-based and algebra-based difficulties starting from extremely simple to complicated ones. a number of strategies at di?erent degrees are offered for certainproblems―thestudentwhoisjustbeginningtolearncalculuswill bene?t from the comparability of the di?erent equipment of resolution. the cloth can also be important for classes in atmospheric physics, environmental points of power iteration and shipping, groundwater hydrology, soil physics,andoceanphysics,andselectedpartsmayevenbeusedforbasic undergraduate physics classes. This selection of routines is predicated on classes taught on the college of Northern British Columbia and on the college of Victoria, Canada.

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1 Differentiation 1 (B) Find an expression for the gradient ∇f (r), where f (r) is a regular function of the radius r ≡ |x| = x2 + y 2 + z 2 . Solution We have ∇f (r) = f ∇r = f =f x y z r, r, r ∂r ∂r ∂r ∂x , ∂y , ∂z =f x r, where f ≡ df /dr. 2 (B) Compute ∇f and the directional derivative ∂f /∂s, where f (x, y) = x sin x2 + y 2 and s = √1 5 (1, 2). 4) 38 EXERCISES IN ENVIRONMENTAL PHYSICS hence the gradient of f is ∇f = = ∂f ∂f , ∂x ∂y sin x2 + y 2 + 2x2 cos x2 + y 2 , 2xy cos x2 + y 2 . The directional derivative of f along s is ∂f ∂s ≡ s · ∇f = · = 1 2 √ ,√ 5 5 sin x2 + y 2 + 2x2 cos x2 + y 2 , 2xy cos x2 + y 2 1 √ sin x2 + y 2 + 2x (x + 2y) cos x2 + y 2 5 .

The solution is defined on (−∞, 0)∪(0, +∞) and x = 0 is a (regular) singular point. We look for power-law solutions y(x) = xr ; substitution of this form into the ODE yields the algebraic equation for the power r r2 + 2r − 1 = 0, √ √ 2−1 2. The solutions y (x) = x with real distinct roots r = −1 ± 1 √ −(1+ 2) are linearly independent and the general soluand y2 (x) = x tion of the ODE on (−∞, 0) ∪ (0, +∞) is therefore √ y(x) = C1 |x| 2−1 √ + C2 |x|−( 2+1) . 9 (B) Solve the linear ODE x2 y + 3xy + y = 0.

3 The graph of f (x) = x2 (1 + λx) for λ = 2. Ordinary differential equations Ordinary differential equations (ODEs) describe physical systems with a finite number of degrees of freedom. Often the solution of partial differential equations, which describe systems with an infinite number of degrees of freedom, can be reduced to the problem of solving a set of ODEs. Many so called zero-dimensional models or box models in earth sciences or environmental sciences neglect the spatial variation of the relevant quantities and retain only their temporal variation: the result is a simplified model of environmental processes based on ODEs that admit analytical solutions.

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