Waveletsch4_5_6_7

Waveletsch4_5_6_7 - Integral Transforms p. 1/93 Integral...

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Unformatted text preview: Integral Transforms p. 1/93 Integral Transforms Math 2025 Gestur lafsson Mathematics Department Louisiana State University Linear Maps Linear Maps Two Important Examples The Integral(cont.) Two Important Examples(cont.) Definition Lemma Examples Examples(cont.) Examples(cont.) Examples(cont.) Examples(cont.) Lemma Counterexample Examples Examples(cont.) Kernel Theorem Remark Examples Examples(cont.) Examples(cont.) Theorem ection 1.2: Linear Maps Integral Transforms p. 2/93 Chapter 4 Linear Maps Linear Maps Linear Maps Two Important Examples The Integral(cont.) Two Important Examples(cont.) Definition Lemma Examples Examples(cont.) Examples(cont.) Examples(cont.) Examples(cont.) Lemma Counterexample Examples Examples(cont.) Kernel Theorem Remark Examples Examples(cont.) Examples(cont.) Theorem ection 1.2: Linear Maps Integral Transforms p. 3/93 Linear Maps We have all seen linear maps before. In fact, most of the maps we have been using in Calculus are linear. Linear Maps Linear Maps Two Important Examples The Integral(cont.) Two Important Examples(cont.) Definition Lemma Examples Examples(cont.) Examples(cont.) Examples(cont.) Examples(cont.) Lemma Counterexample Examples Examples(cont.) Kernel Theorem Remark Examples Examples(cont.) Examples(cont.) Theorem ection 1.2: Linear Maps Integral Transforms p. 4/93 Two Important Examples Example. The Integral To integrate the function f ( x ) = x 2 + 3 x- cosx over the interval [ a, b ] , we first find the antiderivative of x 2 , that is 1 3 x 3 , then the antiderivative of x , which is 1 2 x 2 , and then multiply that by 3 to get 3 2 x 2 . Finally, we find the antiderivative of cosx , which is sinx , and then multiply that by- 1 to get- sinx . To finish the problem we insert the endpoints. Thus, Linear Maps Linear Maps Two Important Examples The Integral(cont.) Two Important Examples(cont.) Definition Lemma Examples Examples(cont.) Examples(cont.) Examples(cont.) Examples(cont.) Lemma Counterexample Examples Examples(cont.) Kernel Theorem Remark Examples Examples(cont.) Examples(cont.) Theorem ection 1.2: Linear Maps Integral Transforms p. 4/93 Two Important Examples Example. The Integral To integrate the function f ( x ) = x 2 + 3 x- cosx over the interval [ a, b ] , we first find the antiderivative of x 2 , that is 1 3 x 3 , then the antiderivative of x , which is 1 2 x 2 , and then multiply that by 3 to get 3 2 x 2 . Finally, we find the antiderivative of cosx , which is sinx , and then multiply that by- 1 to get- sinx . To finish the problem we insert the endpoints. Thus, Z 1- 1 x 2 + 3 x- cosx dx = Z 1- 1 x 2 dx + 3 Z 1- 1 x dx- Z 1- 1 cosx dx = 1 3 x 3 1- 1 + 3 2 x 2 1- 1- [ sinx ] 1- 1 = 2 3- sin 1 + sin (- 1) . Linear Maps Linear Maps Two Important Examples The Integral(cont.) Two Important Examples(cont.) Definition Lemma Examples Examples(cont.) Examples(cont.) Examples(cont.) Examples(cont.) Lemma Counterexample Examples Examples(cont.) Kernel Theorem Remark Examples Examples(cont.)Examples(cont....
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Waveletsch4_5_6_7 - Integral Transforms p. 1/93 Integral...

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