Lagrange multipliers, also called Lagrangian multipliers (e.g., Arfken 1985, p. 945), can be used to find the extrema of a multivariate function
subject to the constraint 
, where 
and 
are functions with continuous first partial
derivatives on the open set containing the curve
,
and 
at any point on the curve (where 
is the gradient).
For an extremum of 
to exist on 
, the gradient of 
must line up with the gradient
of 
.
In the illustration above, 
is shown in red, 
in blue, and the intersection of 
and 
is indicated in light blue. The gradient is a horizontal vector
(i.e., it has no 
-component)
that shows the direction that the function increases; for 
it is perpendicular to the curve, which is a straight line
in this case. If the two gradients are in the same direction, then one is a multiple
(
)
of the other, so
 |
(1)
|
The two vectors are equal, so all of their components are as well, giving
 |
(2)
|
for all 
,
..., 
,
where the constant 
is called the Lagrange multiplier.
The extremum is then found by solving the 
equations in 
unknowns, which is done without inverting 
, which is why Lagrange multipliers can be so useful.
For multiple constraints 
, 
, ...,
 |
(3)
|
