103-102 CALCULUS II nª ¸É 2jessada/CALII/slide/cal_II_5.pdf103-102 CALCULUS II nª ¸É 2 Á...
Transcript of 103-102 CALCULUS II nª ¸É 2jessada/CALII/slide/cal_II_5.pdf103-102 CALCULUS II nª ¸É 2 Á...
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CALCULUS IIThree dimensional spaceScalar Product and Vector ProductLines and Planes in spaceVector-Valued Functions and Space curvesArc Length and the Unit Tangent VectorFunctions of many variablesPartial Derivatives and Chain RuleDirectional Derivatives,Gradient Vectors and Tangent PlanesExtreme values and Saddle Points
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1.��µ¦�εÁ�·��µ¦�ª���Áª�Á�°¦r
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2.��µ¦�εÁ�·��µ¦�¼�Áª�Á�°¦r�oª¥Á�¨µ¦r(scalar) (®¦º°�ªÁ¨�)®¤µ¥Á®�»�®¨��µ��εÁ�·��µ¦�¼�Áª�Á�°¦r�oª¥Á�¨µ¦rÁ¦µ¥´���Å�oÁª�Á�°¦r2.1��¼��oª¥Á�¨µ¦r�ª��Å�o�·«�µ���Á�·¤�Â�n��µ��°�Áª�Á�°¦rÁ�¨É¥�Å�
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u = vÁ¤ºÉ°��Á�}��scalar
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vw v+w
-w
v
w
v
-w
v+(-w)=v-w
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v-v = v+(-v)=
v -v
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�µ¦¡·�µ¦�µÁª�Á�°¦rÄ�¦¼�nª��¦³�°�(Considering vector in component form)
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ijk
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k �º°Áª�Á�°¦r�¹É�¤¸��µ�Á�nµ���1�®�nª¥Â¨³¤¸�·«�µ�Ä�Â�ªÂ���z �µ��ª�
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v=<v1, v2, v3>
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�§¬�¸����oµ�v=<v1, v2, v3>�¨³�w= <w1, w2, w3> ¨oªv+w = < v1+w1, v2+w2, v3+w3>
= (v1+w1)i+(v2+w2)j+(v3+w3)kv-w = < v1-w1, v2-w2, v3-w3>
= (v1-w1)i+(v2-w2)j+(v3-w3)kv = < v1, v2, v3>
= ( v1)i+( v2)j+( v3)k
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�´ª°¥nµ���oµ�v=<-2,1,0>�¨³�w= <3,-4,-5> ¨oªv+w =0.5v =-2w =2w-3v =
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�µ¦®µÁª�Á�°¦r�¹É�¤¸�·«�µ�Á�¸¥ª�´��·«�µ��µ��»��P1�Å�¥´��»��P2�¨³¤¸��µ�Á�nµ�´��´�¦³¥³�µ�¦³®ªnµ��»��P1 ¨³�»��P2
1OP
2OP
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´�Á��ªnµ1 1 2 2OP PP OP
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P1(x1, y1, z1)P2(x2, y2, z2)
2 1 2 1 2 1, ,x x y y z z
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1 2PP
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1 2PP
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�µ¦®µÁª�Á�°¦r�¹É�¤¸�·«�µ�Á�¸¥ª�´��·«�µ��µ��»��P1(0,-2,5) Å�¥´��»��P2 (3,4,-1)�¨³¤¸��µ�Á�nµ�´��´�¦³¥³�µ�¦³®ªnµ��»��P1 (0,-2,5)�¨³�»��P2(3,4,-1)
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�§¬�¸����oµ�v=<v1, v2, v3> ��µ��°��v �º°
2 2 2
1 2 3v v v v
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v
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�§¬�¸����oµ�v=<v1, v2, v3> ��µ��°��v �º°
2 2 2
1 2 3v v v v
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1 2 3v v v v v
2 2 2 2
1 2 3v v v v
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��®µ��µ��°��v= <3,-4,-5>
w= 3i+4j+5k
2v
-3w0
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ªnµ���µ¦�εĮoÁ�}��¦¦���µ�� (normalization)
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