By David V. Widder
Vintage textual content leads from undemanding calculus into extra theoretic difficulties. specified procedure with definitions, theorems, proofs, examples and workouts. subject matters comprise partial differentiation, vectors, differential geometry, Stieltjes necessary, limitless sequence, gamma functionality, Fourier sequence, Laplace rework, even more. quite a few graded workouts with chosen solutions.
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EXAMPLE H. f(x, y) = x when | y | < | x |, f(x, y) −x when | y | | x |. Here f1(0, 0) = 1, f2(0, 0) = 0. Now if f(x, y) were differentiable at (0, 0), equation (5) would become when Δy = Δx But this is a contradiction, as one sees by canceling Δx and letting Δx → 0. Hence f(x, y) is not differentiable at (0, 0). It is continuous there. This function is differentiable at (0, 0) (see Exercise 20). It does not belong to C1 there. To prove this, it is enough to show that f1(x, x) has no limit as x → 0+.
8. Same problem for . 9. If f(x, y) is homogeneous of degree n, show that x2f11 + xyf12 + xyf21 + y2f22 = n(n − 1)f What continuity assumption are you making? 10. Show that when f(x, y) is homogeneous of degree n any derivative of order k is homogeneous of degree n − k. 11. Find f″(t), if f = ex sin y, x = t2, y = 1 − t2, first by the method of the text, then by eliminating x and y before differentiation. 12. 13. 14. If for all positive values of λ u(λx, λy, z/λ) = u(x, y, z) prove xu1 + yu2 = zu2.
Answer the question for each x. �3. Functions of Several Variables We now proceed with a systematic treatment of partial differentiation. We develop first the method of differentiating composite functions analogous to for functions of one variable. 1 LIMITS AND CONTINUITY We begin by defining the limit of a function of two variables. A function f(x, y) approaches a limit A as x approaches a and y approaches b, if, and only if, for each positive number ∈ there is another, δ, such that whenever | x − a | < δ, | y − b | < δ, 0 < (x − a)2 + (y − b)2 we have | f(x, y) − A | < ∈.