CALCULUS 3 FINAL EXAM/ Mul variable Calculus Final
Exam With complete solu on RATED A+ 20254/2026
NEW!! 100% VERIFIED CORRECT
Conserva ve check df/dy=dg/dx
Flux Triple integral (div)
Div <Fx,Fy,Fz>
Stokes' Thmx (Curl F) * r'(t)
Curl ∆
Hyperboloid of one sheet x^2/a^2 + y^2/b^2 - z^2/c^2 = 1
Ellipse x^2/a^2 + y^2/b^2 = z
Ellipsoid x^2/a^2 + y^2/b^2 + z^2/c^2 = 1
Hyperbolic Paraboloid (saddle) z = x^2/a^2 - y^2/b^2
Ellip c Cones x^2/a^2 + y^2/b^2 = z^2/c^2
Equa on of sphere (x-a)^2 + (y-b)^2 + (z-c)^2 = r^2
, Equa on of ball (x-a)^2 + (y-b)^2 + (z-c)^2 ≤ r^2
Area |U x V|
ProjvU ((u•v)/(v•v))v
ScaluV (u•v)/|v|
work |F||d|cosø = F•d
T(t) = Unit tangent Vector r'(t)/|r'(t)|
Principal Unit Normal Vector = N T'/|T'|
Length integral from a to b of |r'(t)|
k = Curvature = (1/|v|)*|dT/dt|= |a X v| / | v^3 |
direc on of vector PQ/|PQ|
Area of triangle (1/2) |u x v|
Gradient <Fx,Fy>
Gradient @ point direc on steepest ascending
Exam With complete solu on RATED A+ 20254/2026
NEW!! 100% VERIFIED CORRECT
Conserva ve check df/dy=dg/dx
Flux Triple integral (div)
Div <Fx,Fy,Fz>
Stokes' Thmx (Curl F) * r'(t)
Curl ∆
Hyperboloid of one sheet x^2/a^2 + y^2/b^2 - z^2/c^2 = 1
Ellipse x^2/a^2 + y^2/b^2 = z
Ellipsoid x^2/a^2 + y^2/b^2 + z^2/c^2 = 1
Hyperbolic Paraboloid (saddle) z = x^2/a^2 - y^2/b^2
Ellip c Cones x^2/a^2 + y^2/b^2 = z^2/c^2
Equa on of sphere (x-a)^2 + (y-b)^2 + (z-c)^2 = r^2
, Equa on of ball (x-a)^2 + (y-b)^2 + (z-c)^2 ≤ r^2
Area |U x V|
ProjvU ((u•v)/(v•v))v
ScaluV (u•v)/|v|
work |F||d|cosø = F•d
T(t) = Unit tangent Vector r'(t)/|r'(t)|
Principal Unit Normal Vector = N T'/|T'|
Length integral from a to b of |r'(t)|
k = Curvature = (1/|v|)*|dT/dt|= |a X v| / | v^3 |
direc on of vector PQ/|PQ|
Area of triangle (1/2) |u x v|
Gradient <Fx,Fy>
Gradient @ point direc on steepest ascending
