By Robert Weinstock
Ebook primarily divided into elements. Chapters 1-4 contain history fabric, easy theorems and isoperimetric difficulties. Chapters 5-12 are dedicated to purposes, geometrical optics, particle dynamics, the idea of elasticity, electrostatics, quantum mechanics and different themes. workouts in every one bankruptcy.
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Additional info for Calculus of Variations with Applications to Physics & Engineering
Domain of product of two functions f (x) and g (x) af af g (x) = dom f x ⋅ g x = dom (f (x)) ∩ dom (g (x)). 3. , = dom the domain of a rational function or the quotient function is the set of all real numbers with the exception of those real numbers for which the function in denominator becomes zero. Notes: 1. , f (x)). 2. (i) The statement “f (x) is defined for all x” signifies a f that f (x) is defined in the interval −∞ , ∞ . (ii) The statements “f (x) is defined in an interval finite or infinite” signifies that f (x) exists and is real for all real values of x belonging to the interval.
Domain = [–5, –1] 2 Now, − x + 5x − 6 ≥ 0 2 ⇒ x − 5x + 6 ≤ 0 a ⇒ x−2 2 −16 x − 24 x Solution: y = a1 − xf a x + 3f Solution: y = a1 − x f a x + 3f Now, a1 − x f a x + 3f ≥ 0 ⇒ − a x − 1f a x + 3f ≥ 0 ⇒ a x − 1f a x + 3f ≤ 0 ⇒ a x − 1f b x − a−3fg ≤ 0 7. y = f a x − 3f ≤ 0 ⇒ x lies between 2 and 3 ⇒2≤x≤3 ∴ domain = [2, 3] 5. y = f a f ⇒ a x + 5f a x + 1f ≤ 0 ⇒ b x − a−1fg b x − a−5fg ≤ 0 ⇒ x lies between –5 ⇒x x+5 + x+5 ≤0 2 4. y = 2 −16 x − 24 x ⇒ −3≤ x ≤1 ∴ domain = [–3, 1] 2 Now, −16 x − 24 x ≥ 0 2 ⇒ − 2 x − 3x ≥ 0 8.
Y = f 1 2 − sin 3 x Solution: Putting 2 – sin 3x ⇒ sin 3x = 2 which is not true for any real x af a ∴ D y = R = −∞ , ∞ n f cos θ = cos α ⇔ θ = 2 n π ± α , n ∈ I Domain of Inverse Trigonometric Functions U| cos θ = cos α V ⇒ θ = n π ± α tan θ = tan α | |W Before studying the method of finding the domain of inverse trigonometric (or, arc) functions, we discuss the domain on which each trigonometric functions is reversible. 2 2 2 2 2 2 sin θ = sin α 3. a ⇒ cos x = − 1 ⇒ x = 2 n + 1 π Notes: 1. If no real solution is available after putting the functional value in denominator = 0, then domain of trigonometric rational functions is R = −∞ , + ∞ .
Calculus of Variations with Applications to Physics & Engineering by Robert Weinstock