By T. J. I'a; Bromwich

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0,0) 0). The point (0, 0) is not in the domain of fI ((x, x, y), so the function has continuous partial derivatives at 1. all points of its domain and the function, by definition, is C Cl. 1 Goals 1. Be able to state the chain rule. 2. Be able to compute a partial derivative by using the chain rule. 3. Be able to compute a tangent vector for a given curve. 2 33 Study Hints 1. Chain rule. Suppose f is a function of Yl, Yl , Y2, ... _+ = - · - + - · - + ·... " However, beware that the "sum" on the right-hand side is df / dx, not n times df /dx.

Higher order partial derivatives. Examples of higher order derivatives are second and third derivatives. With several variables, higher order derivatives may be taken with respect to different variables. The notation 822 ff/8x8y j8x8y means 8(8! 8(8/ 8(8f j8y)j8x, /8y)/8x, which is also denoted fyx· /yx, fyx. 2. Equality of 0/mixed partials. If a function is twice differentiable, then the mixed second partial derivatives may be computed in any order. 3. Warning. Note that the theorem on equality of mixed partials requires continuous partial derivatives.

X(x - Xo) = 0. O. 3 1. Let z Solutions to Every Other Odd Exercise 2y2 2y22 -= F(x,y) = x 22 + 2y o. O. Then 3 = 0. dy dx azjax 8z/8x - azjay· 8z/8y· 8z/8x = 2x and 8zj8y 8z/8y = 4y, so dyjdx dy/dx In this particular case, 8zj8x -2x/4y = -xj2y. -x/2y. Alternatively, one could use the chain rule to get -2xj4y dF/dx = 2x + 4y(dyjdx) 4y(dy/dx) = 0. o. O. Then, rearrangement of the equation dFjdx dy/dx = -xj2y. -x/2y. yields the same result, dyjdx eX eX = 0, we know that dyjdx dy/dx = -(8zf8x)f -(8z/8x)/ 5.

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Quadratic forms and their classification by means of invariant-factors by T. J. I'a; Bromwich

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