By Valerio Faraoni
This can be the 1st ebook in particular dedicated to routines at the software of physics to explain the surroundings together with human influence on it. it's a worthwhile device for college students to enhance abilities within the manipulation of actual ideas and strategies whereas studying environmental technological know-how. The routines are drawn from the author's educating event and the necessity for exciting perform difficulties in numerous environmental physics classes. A bankruptcy on mathematical equipment utilized in the e-book vitamins the fabric.
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Extra resources for Exercises in Environmental Physics (2006)(en)(330s)
A scalar ﬁeld is a scalar function f (x) of the position x = (x, y, z), while a vector ﬁeld is a vector quantity that depends on position, a = a (x). Solution 2 (level B) A mathematically more precise deﬁnition of scalars and vectors can be given by using the transformation properties of their components under a coordinate transformation i i xi −→ x = x (xj ). , s = s, while a covariant vector a with components ai transforms according to ∂x i j i ai −→ a = a . ∂xj j A 1-form or contravariant vector with components ωi instead transforms according to ωi −→ ω i = l ∂xl ωl .
1 (B) a) The population, deﬁned as the number of live individuals P (t), of a certain species in the presence of unlimited food and in the absence of predators or competing species, is described by the Malthus model dP = aP, dt where the birth rate a is constant. Find the future evolution of the species population. b) A better model takes into account the fact that, as the population grows, its members begin competing between themselves for food or other resources, or get poisoned by their own waste products, and the growth cannot continue indeﬁnitely at the same rate.
Dx fy (x, y(x)) x cos (y(x)) + 1 7 (B) Prove that the surfaces of constant f and g, where f (x) = x2 − y 2 , g(x) = xy + C (with C a constant), are orthogonal to each other. Solution The functions f and g are deﬁned and diﬀerentiable at any point (x, y) in the plane. The surfaces of constant f and g are orthogonal if and only if the gradients ∇f and ∇g are mutually perpendicular. We have ∂f ∂f ∇f ≡ = (2x, −2y) , , ∂x ∂y ∇g ≡ ∂g ∂g , ∂x ∂y = (y, x) , and ∇f · ∇g = 2xy − 2yx = 0. The two surfaces are mutually orthogonal at every point (x, y) in the plane.
Exercises in Environmental Physics (2006)(en)(330s) by Valerio Faraoni