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Unformatted text preview: 13. Let be deﬁned by if if Determine if and exist, and if so ﬁnd their values. into the equation, so we use the deﬁnition of partial S OLUTION. In this case, we cannot substitute derivatives: E XERCISE 7. Calculate (a) (b) and . if if at (if they exist) for the following functions. 9 WATERLOO SOS E XAM AID: MATH237 Consider . There are two ﬁrst partial derivatives, and . Each of these are functions of again, so we may take their partial derivatives, to obtain the second partial derivatives of . Thus, we get four possible second derivatives of : We have subscript and operator notation in this case as well, for ease of notation: Remember to not get the order of the variables confused in the various notations. Subscript notation reads left to right (differentiate with respect to the left variable ﬁrst, then the right), while operator and standard notation read right to left (differentiate with respect to the right variable ﬁrst, then the left). We can write out all of the second partial derivatives of a scalar function on in a matrix. D EFINITION 6.2. Let . The Hessian matrix of , denoted by , is deﬁned as E XAMPLE 14. Let of . be deﬁned by . Find all the second partial derivatives S OLUTION. By differentiation rules for single variable (recall example 12), we have Then applying differentiation rules a second time yields You may have noticed that in the previous example, . A natural question is to ask if this always holds; in fact this is not always true, but we can get a weaker condition. 10 T HEOREM 6.3. Let , then . If . and WATERLOO SOS E XAM AID: MATH237 are deﬁned in some neighborhood of and are continuous at has eight We can generalize these ideas to higherorder partial derivatives. For example, third partial derivatives, It turns out that continuous partial derivatives is a very nice property. We introduce some notation in this case. D EFINITION 6.4. Let . If the th partial derivatives of are continuous, then we write and say is in class . For example, if and , then has continuous second partial derivatives. Notice by Theorem 6.3 this implies . 7 Linear Approximations (4.34.4) Having discussed partial derivatives, we are well on our way to deﬁning differentiability in multiple variables. Recall that in 1dimension, the “tangent line” can be used to approximate the graph of a function near the point of tangency. The equation of the tangent line to at the point is given by This equation deﬁnes a function via of at , since it is a linear equation that approximates for sufﬁciently close to . D EFINITION 7.1. Analogously, for at , called the linear approximation . We write this approximation by we deﬁne the linear approximation of at by This plane approximates near . We write this as for sufﬁciently close to . This is also known as the tangent plane to and the point . You may notice we are implicitly assuming the partial derivatives exist; we will develop a more rigorous deﬁnition of the tangent plane in a later section. E XAMPLE 15. Deﬁne . . Find the tangent plane (linear approximation) at the point S OLUTION. By standard differentiation rules, or Then the tangent plane is given by Note: it is helpful to brush up on your trigonometric functions. 11 WATERLOO SOS E XAM AID: MATH237 E XERCISE 8. Verify each approximation (remember, don’t immediately try to expand the terms): (a) (b) (c) , for , for sufﬁciently close to . . . sufﬁciently close to , for sufﬁciently close to E XAMPLE 16. Calculate S OLUTION. Notice and approximately (Use . ). Compare with the calculator value. R EMARK 7.2. This may seem arbitrary at ﬁ...
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This note was uploaded on 11/30/2010 for the course MATH 235/237 taught by Professor Wilkie during the Spring '10 term at Waterloo.
 Spring '10
 WILKIE
 Scalar, Sets

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