multivariate calculus
-
equations & Plane
O
V ECT OR S
. of lines
· . . . . . .
notation : = =
(x , y z) ,
= xi + yj + zk equation of line : (t) (Xo , Yo zo]
=
,
+ + ·
(x0 , Yoz0)
↑
(x2 + y2 + zz
⑳
magnitude (length) : (t) =
equation of plane : ((X-Xo) (y-yo) (z-zo)) n
, ,
· =
0 ↑
(X0 , yo , z0)
1a VECTOR RULES ↳ QUADraTIC SURFACES
↳
directed rector from Po <Xo , Yo zo) =
,
to Pi [X , Y 1, zi] :
=
·
ellipsoid : 4= ++757/
elliptic paraboloid :
III
i = PP =
Pi Po (X,- = -
X0 , Y1 yo , z -
z0) ·
dot product of a =
(ax My az) and , ,
J =
(bx , by bz7
,
: ·
hyperbolic paraboloid (saddle): -
a -
5 axbx + ayby + azbz
=
·
cone : d
a and 5 hyperboloid of 1 sheet :=
n
·
angle between :
9 6
-
:
cosO hyperboloid of 2 sheets :
-
=
191151 ·
projection of 5 along :
proj = S
cross product in R3 :
i5E
x5 =
ax ay az
(aybz-Azby)5 -
(axbz -
azbx)j + (ax by -
aybx)
by by be
↳ x5 is
normal/orthogonal/perpendicular to both a <5
↳ areas of shapes :
partial derivatives
y) 22f
=
1 lim f(x + h , y) f(x
·
-
,
A= laxol fxy fyx =
ey2X22)
=
V
3
· A
lim f(x , y + h) -
f f(X , y) 22f
; fyy 2y2
=
fxx
-
=
H 2x2
· v =
/ :
(bX)/
VECTOR vALUE FUNCTIONS ↳
equation of tangent plane to graph of z = f(x Y), at (Xo , Yo · zo) :
r(t) = (f(t) , g(t) , h(t)) z -
zo = fx (Xo Yo)(X Xo)
,
-
+ fy(X0 yo) <y Yo)
,
-
*
tangent vector : v'(t) differential of w =
f(X , Y , z) :
unit tangent rector : F(t) = -
dw =
df = + dy + dz
ex
T'(t)
unit normal vector : NCt) =
IT'(t) Linear approximation :
((X, y) =
f(a b) , + fx(a b)(X a) ,
-
+ fy(a b)(y b)
,
-
↳
(a , b)
>
-
L(X , y) = f(X , y) near
integrals : Chain Rule :
Suct)dt JSf(t)dt Sg(t)dt ShCtdt]
ya
=
, ,
arc length : 1= Srct)(d -
IT'(t)
curvature : k Level curves f(x , y z)
·
Ir(t) : k
=
=
,
position , velocity acceleration ,
·
Limits of functions :
lim
(t) = v'(t) =
v "(t) f(X , y , z) DNE if different approaches
Newton's 2nd Law : F ma = to (a , b) yield different limits
-
equations & Plane
O
V ECT OR S
. of lines
· . . . . . .
notation : = =
(x , y z) ,
= xi + yj + zk equation of line : (t) (Xo , Yo zo]
=
,
+ + ·
(x0 , Yoz0)
↑
(x2 + y2 + zz
⑳
magnitude (length) : (t) =
equation of plane : ((X-Xo) (y-yo) (z-zo)) n
, ,
· =
0 ↑
(X0 , yo , z0)
1a VECTOR RULES ↳ QUADraTIC SURFACES
↳
directed rector from Po <Xo , Yo zo) =
,
to Pi [X , Y 1, zi] :
=
·
ellipsoid : 4= ++757/
elliptic paraboloid :
III
i = PP =
Pi Po (X,- = -
X0 , Y1 yo , z -
z0) ·
dot product of a =
(ax My az) and , ,
J =
(bx , by bz7
,
: ·
hyperbolic paraboloid (saddle): -
a -
5 axbx + ayby + azbz
=
·
cone : d
a and 5 hyperboloid of 1 sheet :=
n
·
angle between :
9 6
-
:
cosO hyperboloid of 2 sheets :
-
=
191151 ·
projection of 5 along :
proj = S
cross product in R3 :
i5E
x5 =
ax ay az
(aybz-Azby)5 -
(axbz -
azbx)j + (ax by -
aybx)
by by be
↳ x5 is
normal/orthogonal/perpendicular to both a <5
↳ areas of shapes :
partial derivatives
y) 22f
=
1 lim f(x + h , y) f(x
·
-
,
A= laxol fxy fyx =
ey2X22)
=
V
3
· A
lim f(x , y + h) -
f f(X , y) 22f
; fyy 2y2
=
fxx
-
=
H 2x2
· v =
/ :
(bX)/
VECTOR vALUE FUNCTIONS ↳
equation of tangent plane to graph of z = f(x Y), at (Xo , Yo · zo) :
r(t) = (f(t) , g(t) , h(t)) z -
zo = fx (Xo Yo)(X Xo)
,
-
+ fy(X0 yo) <y Yo)
,
-
*
tangent vector : v'(t) differential of w =
f(X , Y , z) :
unit tangent rector : F(t) = -
dw =
df = + dy + dz
ex
T'(t)
unit normal vector : NCt) =
IT'(t) Linear approximation :
((X, y) =
f(a b) , + fx(a b)(X a) ,
-
+ fy(a b)(y b)
,
-
↳
(a , b)
>
-
L(X , y) = f(X , y) near
integrals : Chain Rule :
Suct)dt JSf(t)dt Sg(t)dt ShCtdt]
ya
=
, ,
arc length : 1= Srct)(d -
IT'(t)
curvature : k Level curves f(x , y z)
·
Ir(t) : k
=
=
,
position , velocity acceleration ,
·
Limits of functions :
lim
(t) = v'(t) =
v "(t) f(X , y , z) DNE if different approaches
Newton's 2nd Law : F ma = to (a , b) yield different limits
