EC-18.6 Functions of Several Variables, and Partial Derivatives
You can take partial derivatives of any order, read a gradient as the direction of steepest increase, and find the best value of a function of two variables with or without a constraint tying them together.
Before:nothing requiredUnlocks:EC-3. Differential Equations
Nothing in engineering depends on one quantity alone. A gain depends on a resistor and a supply voltage, a field depends on three coordinates, and a loss depends on every parameter at once. This topic is where the derivative learns to handle that, and it is also where a reader coming from the machine learning side of the site meets the same gradient from the other direction.
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Functions of two or more variables, and how to picture one
A function of two variables is a surface, and a function of three is something no drawing captures. Level curves and level surfaces are the practical substitute, and they are how a contour map of any performance measure is read.
NPTEL: Multivariable Calculus (IIT Roorkee) · CourseLimits and continuity when there is more than one way in
Approaching a point in the plane can be done along infinitely many routes, and a limit exists only when every one of them agrees. Showing two routes that disagree is the standard way to prove a limit does not exist.
NPTEL: Multivariable Calculus (IIT Roorkee) · CoursePartial derivatives of first and higher order
A partial derivative measures the change due to one variable while the others are held fixed. Mixed second derivatives taken in either order agree for the functions met in practice, which is a convenience worth knowing is not automatic.
MIT 18.02SC: Multivariable Calculus · CourseDifferentiability, the total differential, and the chain rule in several variables
Having both partial derivatives is weaker than being differentiable, and the total differential is what says how the output moves when every input moves a little. That expression is the basis of every error-propagation calculation in the measurements course.
The gradient, the directional derivative, and steepest increase
Collecting the partial derivatives into one object gives a direction, and that direction is the one in which the function grows fastest. Walking against it is what every optimiser in the machine learning area is doing.
MIT 18.02SC: Multivariable Calculus · CourseUnconstrained maxima and minima, and the Lagrange multiplier
A best value with no restrictions sits where the gradient vanishes, and the second-order test says which kind of point it is. When a constraint ties the variables together, the multiplier method finds the answer without solving the constraint first.
NPTEL: Multivariable Calculus (IIT Roorkee) · Course
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