103-102 CALCULUS II ??????? 2

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103-102 CALCULUS II ??????? 2
????????????????? CALCULUS II ??????? 2
?????????? ????????????? 3 ???? ???? ???????,
?????, ???????? (Vectors) ?? 3 ???? ???
???????????????????????????? ????????????????????
3 ????, ??????????????? 3 ????????????????????????
?
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CALCULUS II
Three dimensional space
Scalar Product and Vector Product
Lines and Planes in space
Vector-Valued Functions and Space curves
Arc Length and the Unit Tangent Vector
Functions of many variables
Partial Derivatives and Chain Rule
Directional Derivatives,
Gradient Vectors and Tangent Planes
Extreme values and Saddle Points
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???????????? (2 ????)
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???????????????????? (????????) Cartesian
(Rectangle) Coordinate
????????? ????? (0,0,0)
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?????? (Quadrant)
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?????? (Octant)
????????????????????? ?????????????
????????? ????????????????? 3 ???? ???? 8 ????
(Oct ??? 8)
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?????????????? (1st Octant)
???????????? (2nd Octant)
???????????? (3rd Octant)
???????????? (4th Octant)
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???????????? (5th Octant)
??????????? (6th Octant)
????????????? (7th Octant)
???????????? (8th Octant)
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????? xy (xy-plane)
????? xy ?????????????????? ????????????? (x,y,0)
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????? xz (xz-plane)
????? xz ?????????????????? ????????????? (x,0,z)
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????? yz (yz-plane)
????? yz ?????????????????? ????????????? (0,y,z)
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??? x (x-axis)
??? x ?????????????????? ????????????? (x,0,0)
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??? y (y-axis)
??? y ?????????????????? ????????????? (0,y,0)
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??? z (z-axis)
??? z ?????????????????? ????????????? (0,0,z)
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???????
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????????????????? 2 ??????
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????????????????? 2 ??????
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????????????????????? (2,3,-1) ??? ??? (4,-1,3)
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????????????????????? (1,1, 3) ??? ??? (-3,-1,-1)
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???????????????????????????? ????????????????
(0,0,0) ??????????? 5 ?????
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???????????????????????????????????????? (0,0,0)
????? ????? 5 ????? ???
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?????????????????????????????????? (x0, y0, z0)
????? ????? r ????? ???
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?????????????????????????????????? (x0, y0, z0)
????? ????? r ????? ???
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?????????????????????????????????? (-1,0,-4)
????? ????? 3 ????? ???
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?????????????????????????????????? (-1,2,-3)
????? ????? 11 ????? ???
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??????????????????????????????????????
??????????????????? ????????????????
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?????????????????
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?????????????????
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????????????????????? 2 ???
?????????????????? M ?????????????????????????????
??????? ?????????? 2 ??? P1(x1, y1, z1) ???
P2(x2, y2, z2) ???????????
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????????????????????????????????????? P1(3,-2,0)
??? P2(7, 4, 4)
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?????????? ???????????????????????? ???
??????? ???????????????????????? ????????????????
???????? ??? ?????????????????????????????????????
??????????? ???????????????????????????????
???????? (Vector)
?????!!!
???????? ?????????????????????????????????????????
?????????
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?????????? ? ????????????????????????????????????
?????????????????????????????????????????????????
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????????????????????????? ??? ????????????????
?????? ???????????????????????
(1,2,1)
(0,1,1)
(1,0,0)
(2,1,0)
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???????? 0 ???????????????????????????? 0
????????????????????????? ??????
??????????????????????? 0 ???? ?????????
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??????????????????????
1. ????????????????????????? 1.1 vw
wv 1.2 0v v0 v ???????? ???????? 2
?????????????? ???????????????
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2. ??????????????????????????????????(scalar)
(??????????) ???????? ????????????????????????????
????????????????????????????? 2.1
????????????????? ???????????????
?????????????????? ????????? 2.1.1
??????????????????????? 2.1.2
???????????????????????
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2.2 ???????????????? ????????????????????
?????????? ????????????????? 2.2.1
??????????????????????? 2.2.2
???????????????????????
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???????? ??????????? u ??? v ?????????????? 0
??????? ???????????? u ??? v ?????????????????
u ?v
????? ? ???? scalar
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???????? ?????????????????????????????????????? ??
??????????? ??????? v-w v(-w) ???????
w
v
w
vw
v
-w
-w
v
v(-w)v-w
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v-v v(-v)
v
-v
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????????????????????????????????? (Considering
vector in component form)
?????????????????????????????????
???????? ????????????? ??????????????????
?????????????????????????????????????????????????
?????????????????????????????????????????????????
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???????? i,j ??? k
i ???????????????????????????? 1 ?????
??????????????????? x ??????
k
j
j ???????????????????????????? 1 ?????
??????????????????? y ??????
i
k ???????????????????????????? 1 ?????
??????????????????? z ??????
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?????????? ????????????????????
??????????????????????????????????????????? i,j
??? k ???????
vv1i v2j v3k
????????????????????????????????? v
????????????????????????
vltv1, v2, v3gt
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??????? ???????? 2 ????????????????????????????? ?
?????????????????? ???????????????????
???? lta,b,cgt lt-2, , gt ??????????
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??????? ??? vltv1, v2, v3gt ??? w ltw1, w2, w3gt
???? vw lt v1w1, v2w2, v3w3gt
(v1w1)i(v2w2)j(v3w3)k v-w lt v1-w1, v2-w2,
v3-w3gt (v1-w1)i(v2-w2)j(v3-w3)k v lt
v1, v2, v3gt ( v1)i( v2)j( v3)k
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???????? ??? vlt-2,1,0gt ??? w lt3,-4,-5gt
???? vw 0.5v -2w 2w-3v
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??????????????????????????????????????? ?????? P1
???????? P2 ??????????????????? ?????????????????
P1 ?????? P2
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?????????
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??????????????????????????????????????? ?????? P1
???????? P2 ??????????????????? ?????????????????
P1 ?????? P2
P2(x2, y2, z2)
P1(x1, y1, z1)
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????????????? 3 ?????????
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??????????????????????????????????????? ??????
P1(0,-2,5) ???????? P2 (3,4,-1)
??????????????????? ????????????????? P1 (0,-2,5)
?????? P2(3,4,-1)
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??????? ?????????????? u,v ??? w ?????????? k
??? l ??? ????
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???????????????
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??????? ??? vltv1, v2, v3gt ??????? v ???
??????? v ???
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??????? ??? vltv1, v2, v3gt ??????? v ???
??????? v ???
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???????????
v lt3,-4,-5gt
w 3i4j5k
2v
-3w
0
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??????????????????
??? v ?????????????? 0 ????????????????? 1 ?????
??? ??????????????????????????? v ???
????????
???????? ?????????????????????????????????
??? ????????????????????? (normalization)
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???????????? 1 ????? ???????????????????????? ????
????
v lt2,-2,1gt