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Summary Calculus III Formula Sheet $5.48
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Summary

Summary Calculus III Formula Sheet
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I wrote this document to compile all the formulas I needed for the Multivariable Calculus (Calc III) class. This list includes vector, integral, derivative, time, Lagrange, and mass formulas, and many more. Hope this helps!

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  • December 28, 2024
  • 5
  • 2023/2024
  • Summary

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1 + (y y ) )2 cylindrical surface
2
(
equation of a
sphere : x + (2 2 = ra :
one var
missing
-
-

-


.

, . ,



( ) center
,, y ,,
2
,
=




Ac + By+ [22 +
(collinear) quadratic surface : Dec + Ey + Fz +
G

parallel rectors := G > O

·

ellipsoid : A , B ,
C >

dot product : B - =
a ,
b ,
+ azbe
·
one-sheet hyperboloid : A , B > 0 ; (0

- = 11/12 ·
two-sheet hyperboloid :
As0 ; B C ,
< O

G =
G
2 5 .




angle between vectors : O A B 0 (O
>
Hallibil
·
cone
=
COS :
,
,




AorBorC = 0
(corollary/ orthogonal)
perpendicular vectors : 5
:
·
elliptic paraboloid (bowl") Acc +
0 z By
: =
.
=




2
·
hyperbolic paraboloid ("saddle") : z =
Ax2-By
orthogonal decomposition : =
pric()
<

rotational surface x + :




y2 =
f(z)

work (w) =
Fo & negative
j T
( y z)
rectangular coordinates :
, ,




cross product : x5 = a ,
a
, as
cylindrical coordinates (r :
, A z) ,


b , bz by
spherical coordinates (S :
,
0 ,
9)


x = -



(ax) R =
C C- R

x = 10 ,
0 ,
0 ·
x = rcOSO ·
r = vxz yz +


·

y
= rsinG ·
tand =
orthogonal vector : x to &
- S 53C
Ilax5 f2 r2 + z3 Scost
of
parallelogram
·
·
area : =
z =




·
tand = ·
r =
Ssin
torque : x &


parallelapiped volume : (BA) A ·
rector-valued line F :


= B + at


tetrohedron : El parallelapiped volume vector-valued circle : v =
Lacost , sint) : Oct

> >
-


coplanar :
(ABAC) .*D =
O helix : =
Lacost ,
asint , t) &

:
E r
canonical equation of line Cycloid :
=
r(t-sint)

r(l- cost)


S
x + at
y
x = =




parametric equation : y =
y .
+ bt
(a ,
b ,
c)
2 =
2. + Ct
Mo(o Yo Zol , , limit of :) =

(f(t) , 9(t) ,(t)
equation of a plane : Ak -() +
B(y yo) - + ((z 2) -
=
0 derivative : '(E) =
(f'(t) g'(t) nltl) , ,




Ax +
By + Cz + D


integral Sr(t)dt (ff(t)dt Sg(t)dt ShIt)dt) Y
, , ,



distance from point to plane : d :
=
+
VA2
=

B2 C
, ,
+ +




Intell
distance from point to line : d =


IIell

, position vector : F(t)


velocity rector : Y(t) = '(t) always find

(t)
TIME
acceleration vector a(t) :
=
u(t) = "(t)



speed : 11 ELIII


projectile motion : () =
((Vosdt , (Vosindt -)

tangent unit vector : F(t) =
i

principle unit vector : (t) =


acceleration decomposition alt :

=
af an
+


⑧ calculate '(t) , "(t)
O calculate
>
-

② a+ = (t) ·
T
>
-
⑧ an =
a(t) -



a +
T


①
an
= Il step 311 : N=



arclength
: s = III at



II"Il
curvature : k =
11/113

Cor)
AN
k =
115-112

(or)

1y"l
k =

(1 +
y(z)5

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