Calculo Vetorial

Integrais de Linha de Funções Escalares 1 Calcule as seguintes integrais curvilíneas de primeira espécie d C y2 ds onde C é o primeiro arco da ciclóide x at sen t y a1 cos t f C x2 y22 ds onde C é o arco da espiral logarítmica r aemθ m 0 desde o ponto A0a até o ponto O 0 g C x y ds onde C é o laço direito da lemniscata r a2 cos2θ Teorema de Green 4 Calcule γ F dr onde F xy 4x3 y3 i 3x4 y2 5x2 j e γ é a fronteira do quadrado de vértices 10 01 10 e 01 Podemos escrever 𝑟𝑡 𝑥𝑡 𝑦𝑡 𝑎𝑡 sin𝑡 𝑎1 cos𝑡 Derivada em relação a t 𝑟𝑡 𝑎 𝑑 𝑑𝑡 𝑡 sin𝑡 𝑎 𝑑 𝑑𝑡 1 cos𝑡 𝑟𝑡 𝑎 1 cos𝑡 𝑎0 sin𝑡 𝑎 1 cos𝑡 sin𝑡 Calculando a norma do vetor derivada 𝑟𝑡 𝑎2 1 cos𝑡2 sin𝑡2 𝑎 1 2 cos𝑡 cos2𝑡 sin2𝑡 cos2𝑡 sin2𝑡 1 𝑟𝑡 𝑎 2 2 cos𝑡 2 𝑎 1 cos𝑡 1 cos𝑡 2 sin2 𝑡 2 𝑟𝑡 2 𝑎 2 sin2 𝑡 2 2𝑎 sin 𝑡 2 Assim 𝑑𝑠 𝑟𝑡𝑑𝑡 2𝑎 sin 𝑡 2 𝑑𝑡 O integrando é 𝑦2 𝑎 1 cos𝑡 2 𝑎2 2 sin2 𝑡 2 2 𝑦2 4𝑎2 sin4 𝑡 2 Então 𝑦2 𝐶 𝑑𝑠 𝑦2 2𝜋 0 𝑑𝑠 4𝑎2 sin4 𝑡 2 2𝜋 0 2𝑎 sin 𝑡 2 𝑑𝑡 8𝑎3 sin5 𝑡 2 𝑑𝑡 2𝜋 0 𝑦2 𝐶 𝑑𝑠 8𝑎3 2 cos 𝑡 2 4 3 cos3 𝑡 2 2 5 cos5 𝑡 2 0 2𝜋 𝑦2 𝐶 𝑑𝑠 8𝑎3 2 2 4 3 4 3 2 5 2 5 0 2𝜋 8𝑎3 32 15 256𝑎3 15 𝑟 𝑎𝑒𝑚𝜃 𝑟2 𝑎2𝑒2𝑚𝜃 𝑥 𝑟 cos𝜃 𝑦 𝑟 sin𝜃 𝑥2 𝑦2 𝑟2 Derivada em relação ao ângulo 𝑟𝜃 𝑑𝑟 𝑑𝜃 𝑎 𝑑 𝑑𝜃 𝑒𝑚𝜃 𝑎𝑚𝑒𝑚𝜃 𝑑𝑠 𝑟2 𝑑𝑟 𝑑𝜃 2 𝑑𝜃 𝑎𝑒𝑚𝜃2 𝑎𝑚𝑒𝑚𝜃2𝑑𝜃 𝑎𝑒𝑚𝜃 1 𝑚2𝑑𝜃 𝑥2 𝑦22 𝑟22 𝑟4 𝑎4𝑒4𝑚𝜃 A integral é 𝑥2 𝑦22 𝐶 𝑑𝑠 𝑎4 𝜋 𝜋 2 𝑒4𝑚𝜃 𝑎𝑒𝑚𝜃 1 𝑚2𝑑𝜃 𝑎5 1 𝑚2 𝑒5𝑚𝜃 𝜋 𝜋 2 𝑑𝜃 𝑥2 𝑦22 𝐶 𝑑𝑠 𝑎5 1 𝑚2 𝑒5𝑚𝜃 5𝑚 𝜋 2 𝜋 𝑎5 1 𝑚2 𝑒5𝑚𝜋 𝑒 5𝑚𝜋 2 5𝑚 𝑥2 𝑦22 𝐶 𝑑𝑠 𝑎5 1 𝑚2 𝑒5𝑚𝜋 𝑒 5𝑚𝜋 2 5𝑚 𝑟 𝑎2 cos2𝜃 𝑥 𝑟 cos𝜃 𝑎2 cos2𝜃 cos𝜃 𝑦 𝑟 sin𝜃 𝑎2 cos2𝜃 sin𝜃 𝑥 𝑦𝑑𝑠 𝐶 1 2 𝑥 𝑦𝑟2 𝜋 4 𝜋 4 𝑑𝜃 𝑥 𝑦𝑟2 𝑎2 cos2𝜃cos𝜃 𝑎2 cos2𝜃 sin𝜃 𝑎4 cos22𝜃 𝑥 𝑦𝑟2 𝑎6cos32𝜃 cos𝜃 cos32𝜃 sin𝜃 𝑥 𝑦𝑑𝑠 𝐶 𝑎6 2 cos32𝜃 cos𝜃 𝑑𝜃 𝜋 4 𝜋 4 cos32𝜃 sin𝜃 𝑑𝜃 𝜋 4 𝜋 4 Vamos usar várias identidades trigonométricas cos32𝜃 cos𝜃 2 cos2𝜃 13 cos𝜃 cos32𝜃 cos𝜃 8 cos6𝜃 12 cos4𝜃 6 cos2𝜃 1 cos𝜃 cos32𝜃 cos𝜃 𝑑𝜃 8 cos7𝜃 12 cos5𝜃 6 cos3𝜃 cos𝜃𝑑𝜃 cos32𝜃 cos𝜃 𝑑𝜃 𝜋 4 𝜋 4 8 7 sin7𝜃 12 5 sin5𝜃 2 sin3𝜃 sin𝜃 𝜋 4 𝜋 4 cos32𝜃 cos𝜃 𝑑𝜃 𝜋 4 𝜋 4 162 35 cos32𝜃 cos𝜃 8 cos6𝜃 12 cos4𝜃 6 cos2𝜃 1 sin𝜃 cos32𝜃 sin𝜃 𝑑𝜃 8 cos6𝜃 12 cos4𝜃 6 cos2𝜃 1sin𝜃 𝑑𝜃 cos32𝜃 cos𝜃 𝑑𝜃 𝜋 4 𝜋 4 8 7 cos7𝜃 12 5 cos5𝜃 2 cos3𝜃 cos𝜃 𝜋 4 𝜋 4 0 𝑥 𝑦𝑑𝑠 𝐶 𝑎6 2 162 35 0 82 35 𝑎6 Teorema de Green 𝑃𝑑𝑥 𝛾 𝑄𝑑𝑦 𝑄 𝑥 𝑃 𝑦 𝑑𝐴 𝑆 𝑃 4𝑥3𝑦3 𝑄 3𝑥4𝑦2 5𝑥2 𝑄 𝑥 23𝑥4𝑦2 5𝑥12𝑦2𝑥3 5 𝑃 𝑦 12𝑥3𝑦2 𝑄 𝑥 𝑃 𝑦 23𝑥4𝑦2 5𝑥12𝑦2𝑥3 5 12𝑥3𝑦2 72𝑥7𝑦4 150𝑥4𝑦2 50𝑥 12𝑥3𝑦2 𝑃𝑑𝑥 𝛾 𝑄𝑑𝑦 𝐹 𝛾 𝑑𝑟 72𝑥7𝑦4 150𝑥4𝑦2 50𝑥 12𝑥3𝑦2𝑑𝐴 𝑆 Vamos calcular a integral na região x0 y0 e multiplicar por 4 𝐹 𝛾 𝑑𝑟 4 72𝑥7𝑦4 150𝑥4𝑦2 50𝑥 12𝑥3𝑦2𝑑𝑦𝑑𝑥 𝑥1 0 1 0 𝐹 𝛾 𝑑𝑟 4 50𝑥𝑦 4𝑥3𝑦3 50𝑥4𝑦3 72𝑥7𝑦5 5 0 𝑥1 1 0 𝑑𝑥 𝐹 𝛾 𝑑𝑟 4 50𝑥𝑥 1 4𝑥3𝑥 13 50𝑥4𝑥 13 1 0 72𝑥7𝑥 15 5 𝑑𝑥 𝐹 𝛾 𝑑𝑟 4 50𝑥3 3 25𝑥2 4𝑥7 7 2𝑥6 12𝑥5 5 𝑥4 0 1 4 50 𝑥8 8 3𝑥7 7 𝑥6 2 𝑥5 5 0 1 4 72𝑥13 65 6𝑥12 144𝑥11 11 72 5 𝑥10 8𝑥9 9𝑥8 5 0 1 𝐹 𝛾 𝑑𝑟 4 17 2 34 Podemos escrever 𝑟 𝑡𝑥 𝑡 𝑦 𝑡 𝑎𝑡 sin𝑡 𝑎1cos 𝑡 Derivada em relação a t 𝑟 𝑡𝑎 𝑑 𝑑𝑡 𝑡 sin 𝑡 𝑎 𝑑 𝑑𝑡 1cos 𝑡 𝑟 𝑡𝑎1cos 𝑡𝑎0sin 𝑡 𝑎1cos 𝑡 sin 𝑡 Calculando a norma do vetor derivada 𝑟 𝑡𝑎 21cos 𝑡 2sin 𝑡 2𝑎 12cos 𝑡 cos 2 𝑡sin 2𝑡 cos 2𝑡 sin 2𝑡 1𝑟 𝑡𝑎22cos 𝑡2𝑎 1cos 𝑡 1cos 𝑡 2sin 2 𝑡 2𝑟 𝑡2 𝑎 2sin 2 𝑡 22𝑎sin 𝑡 2 Assim 𝑑𝑠𝑟 𝑡 𝑑𝑡2𝑎sin 𝑡 2𝑑𝑡 O integrando é 𝑦 2𝑎1cos 𝑡 2𝑎 22sin 2 𝑡 2 2 𝑦 24𝑎 2sin 4 𝑡 2 Então 𝐶 𝑦 2𝑑𝑠 0 2𝜋 𝑦 2𝑑𝑠 0 2𝜋 4 𝑎 2sin 4 𝑡 22𝑎 sin 𝑡 2𝑑𝑡8𝑎 3 0 2 𝜋 sin 5 𝑡 2𝑑𝑡 𝐶 𝑦 2𝑑𝑠8𝑎 32cos 𝑡 2 4 3 cos 3 𝑡 2 2 5 cos 5 𝑡 20 2𝜋 𝐶 𝑦 2𝑑𝑠8𝑎 322 4 3 4 3 2 5 2 50 2𝜋 8 𝑎 3 32 15256𝑎 3 15 𝑟𝑎𝑒 𝑚𝜃𝑟 2𝑎 2𝑒 2𝑚𝜃𝑥𝑟 cos 𝜃 𝑦𝑟 sin 𝜃𝑥 2 𝑦 2𝑟 2 Derivada em relação ao ângulo 𝑟 𝜃 𝑑𝑟 𝑑𝜃𝑎 𝑑 𝑑 𝜃 𝑒 𝑚𝜃𝑎𝑚𝑒 𝑚𝜃 𝑑𝑠 𝑟 2 𝑑𝑟 𝑑 𝜃 2 𝑑 𝜃𝑎 𝑒 𝑚𝜃 2𝑎𝑚𝑒 𝑚𝜃 2𝑑 𝜃𝑎𝑒 𝑚𝜃1𝑚 2𝑑 𝜃 𝑥 2 𝑦 2 2𝑟 2 2𝑟 4𝑎 4𝑒 4𝑚 𝜃 A integral é 𝐶 𝑥 2 𝑦 2 2𝑑𝑠 𝜋 2 𝜋 𝑎 4𝑒 4𝑚𝜃𝑎𝑒 𝑚𝜃1𝑚 2𝑑𝜃𝑎 51𝑚 2 𝜋 2 𝜋 𝑒 5𝑚𝜃 𝑑𝜃 𝐶 𝑥 2 𝑦 2 2𝑑𝑠𝑎 51𝑚 2 𝑒 5𝑚 𝜃 5𝑚 𝜋 2 𝜋 𝑎 51𝑚 2 𝑒 5𝑚 𝜋𝑒 5𝑚 𝜋 2 5𝑚 𝐶 𝑥 2 𝑦 2 2𝑑𝑠𝑎 51𝑚 2 𝑒 5𝑚 𝜋𝑒 5𝑚 𝜋 2 5𝑚 PREPARATION OF MEDICAL READING INTERSECTIONS 1 Read the title of the chapter and the prepared notes 2 Try to find the difficult words and look them up in the dictionary 3 Read the text word by word till you understand the whole text 4 Ask yourself the questions What does this text mainly talk about What is the main idea of this chapter 5 Follow the main idea read the text again and circle the words that are related to the main idea 6 Write the summary of the paragraph using the words circled in the text 7 Write the main terms of the paragraph with their definitions 8 Check your understanding of the paragraph by talking about it addressing these questions Who What When Where Why How 9 Retell the paragraph and make a brief medical reading 10 Link the information you have obtained with your future profession The doctor may examine the heart in the patient with a stethoscope or place the patient in a special bath and send a weak electrical current The doctor usually listens to the heart through the chest to identify abnormal heart beats If the doctor feels a pulse on the wrist it means that the heart is regularly pumping the blood throughout the patients body The doctor may check the pulse at the patients neck groin or the back of the knee It is important for the doctor to examine the patients heart carefully in order to diagnose his illness properly Electrical impulses originating in the heart stimulate the heart to beat regularly LINGVIST The text is mainly about the examination of the heart It provides information about the methods the doctors usually use to examine the heart the importance of diagnosing heart diseases and the electrical impulses stimulating the heart to beat normally The main idea is that the heart in the patient can be examined using different methods and an accurate diagnosis of heart diseases can be made Key words heart examination stethoscope diagnosis abnormal heart beats pulse electrical impulses The heart rate and pulse Wave Factor indicate how much of cardiac output passes through capillaries and veins in peripheral resistant net Which is important for arterial blood pressure regulation There are two types of ordinary and emergency examination of the heart Medical examination is done by palpation percussion auscultation ECG echocardiography applying load tests heart catheterization and xray methods The most important part of medical examination of the heart is auscultation with a stethoscope Normal heart sound murmurs are studied at auscultations Examining the pulse in radial artery is a simple and popular way of cardiology examination The pulse rate pulse strength rhythm and pulse tension are noted carefully The pulse at the carotid artery on each side is felt which may for the diseased arteries located in that region Medical examination of patients heart is very important in cardiology so the medical doctor should be able to make this examination carefully and accurately within limited time It takes some time to get the necessary knowledge and practical skills for this kind of examination which is done daily in cardiology The text mainly tells about the importance of medical examination of heart and methods of examination Key words medical examination the heart pulse auscultation ECG echocardiography palpation percussion murmur catheterization cardiology 𝑟𝑎 2cos2𝜃 𝑥𝑟 cos 𝜃 𝑎 2cos 2𝜃cos𝜃 𝑦𝑟 sin 𝜃 𝑎 2cos 2 𝜃sin 𝜃 𝐶 𝑥𝑦 𝑑𝑠1 2 𝜋 4 𝜋 4 𝑥 𝑦 𝑟 2𝑑𝜃 𝑥 𝑦 𝑟 2𝑎 2cos 2𝜃cos 𝜃𝑎 2cos 2 𝜃sin 𝜃𝑎 4cos 2 2 𝜃 𝑥 𝑦 𝑟 2𝑎 6cos 3 2𝜃cos 𝜃cos 32𝜃 sin 𝜃 𝐶 𝑥𝑦 𝑑𝑠𝑎 6 2 𝜋 4 𝜋 4 cos 32 𝜃cos𝜃 𝑑𝜃 𝜋 4 𝜋 4 cos 3 2𝜃sin 𝜃𝑑 𝜃 Vamos usar várias identidades trigonométricas cos 32 𝜃cos 𝜃 2cos 2𝜃1 3cos 𝜃 cos 32 𝜃cos 𝜃 8cos 6 𝜃12cos 4 𝜃6cos 2 𝜃1cos 𝜃 cos 3 2𝜃cos𝜃𝑑𝜃 8cos 7 𝜃12cos 5𝜃6cos 3𝜃cos𝜃𝑑 𝜃 𝜋 4 𝜋 4 cos 32𝜃 cos 𝜃 𝑑 𝜃 8 7sin 7𝜃 12 5 sin 5 𝜃 2sin 3𝜃sin 𝜃 𝜋 4 𝜋 4 𝜋 4 𝜋 4 cos 32𝜃 cos 𝜃 𝑑 𝜃162 35 cos 32 𝜃cos 𝜃 8cos 6 𝜃12cos 4 𝜃6cos 2 𝜃1sin 𝜃 cos 3 2𝜃sin 𝜃 𝑑 𝜃 8cos 6𝜃12cos 4𝜃6cos 2𝜃1sin 𝜃 𝑑𝜃 𝜋 4 𝜋 4 cos 32𝜃 cos 𝜃 𝑑 𝜃 8 7 cos 7 𝜃 12 5 cos 5 𝜃2cos 3𝜃cos 𝜃 𝜋 4 𝜋 4 0 𝐶 𝑥𝑦 𝑑𝑠𝑎 6 2 162 35 082 35 𝑎 6 Teorema de Green 𝛾 𝑃𝑑𝑥𝑄𝑑𝑦 𝑆 𝑄 𝑥 𝑃 𝑦 𝑑𝐴 𝑃4 𝑥 3 𝑦 3𝑄3𝑥 4 𝑦 25𝑥 2 𝑄 𝑥 2 3𝑥 4 𝑦 25 𝑥 12 𝑦 2𝑥 35 𝑃 𝑦 12𝑥 3 𝑦 2 𝑄 𝑥 𝑃 𝑦 2 3𝑥 4 𝑦 25 𝑥 12 𝑦 2𝑥 3512 𝑥 3 𝑦 272 𝑥 7 𝑦 4150𝑥 4 𝑦 250 𝑥12 𝑥 3 𝑦 2 𝛾 𝑃𝑑𝑥𝑄𝑑𝑦 𝛾 𝐹 𝑑𝑟 𝑆 72𝑥 7𝑦 4150𝑥 4 𝑦 250 𝑥12 𝑥 3 𝑦 2𝑑𝐴 Vamos calcular a integral na região x0 y0 e multiplicar por 4 𝛾 𝐹 𝑑𝑟4 0 1 0 𝑥1 72 𝑥 7 𝑦 4150𝑥 4 𝑦 250 𝑥12 𝑥 3 𝑦 2𝑑𝑦𝑑𝑥 𝛾 𝐹 𝑑𝑟4 0 1 50 𝑥𝑦 4 𝑥 3 𝑦 350𝑥 4 𝑦 3 72𝑥 7 𝑦 5 5 0 𝑥 1 𝑑𝑥 𝛾 𝐹 𝑑𝑟4 0 1 50𝑥 𝑥14𝑥 3 𝑥1 350 𝑥 4 𝑥1 3 72𝑥 7𝑥1 5 5 𝑑𝑥 𝛾 𝐹 𝑑𝑟4 50 𝑥 3 3 25𝑥 2 4 𝑥 7 7 2𝑥 6 12𝑥 5 5 𝑥 40 1 450 𝑥 8 8 3 𝑥 7 7 𝑥 6 2 𝑥 5 5 0 1 4 72𝑥 13 65 6 𝑥 12 144 𝑥 11 11 72 5 𝑥 10 8 𝑥 9 9 𝑥 8 5 0 1 𝛾 𝐹 𝑑𝑟 4 17 2 34