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Partial derivatives

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Recall

Definition. A vector $\vec{AB}$ is a geometric object or quantity that has both magnitude (or length) and direction. Vectors in an n-dimensional Euclidean space can be represented as coordinates vectors in a Cartesian coordinate system.

Definition. The magnitude of the vector $\vec{A}$, also known as length or norm, is given by the square root of the sum of its components squared, $|\vec{A}|~ or~ ||\vec{A}|| = \sqrt{a_1^2+a_2^2+a_3^2}$, e.g., $||< 3, 2, 1 >|| = \sqrt{3^2+2^2+1^2}=\sqrt{14}$, $||< 3, -4, 5 >|| = \sqrt{3^2+(-4)^2+5^2}=\sqrt{50}=5\sqrt{2}$, or $||< 1, 0, 0 >|| = \sqrt{1^2+0^2+0^2}=\sqrt{1}=1$.

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Partial derivatives

Partial derivatives are an extension of derivatives to functions of multiple variables. They measure how a function changes as one of its variables changes, while keeping the other variables constant. This concept is fundamental in multivariable calculus and has various applications in fields like physics, engineering, and economics.

Definition

Consider a function f(x1, x2, ···, xn) that depends on n variables. The partial derivative of f with respect to one of these variables, say xi, is the derivative of f while holding the other variables constant.

Formal Definition

For a function f(x,y), the partial derivative with respect to x at a point (x0, y0) is defined as:

$\frac{\partial f}{\partial x}(x_0, y_0) = \lim_{\Delta x \to 0} \frac{f(x_0+\Delta x, y_0)-f(x_0, y_0)}{\Delta x}$

Using a different notation, $\frac{\partial f}{\partial x}(x, y) = \lim_{h \to 0} \frac{f(x+h, y)-f(x, y)}{h}$

Similarly, the partial derivative with respect to y at a point (x0, y0) is defined as: $\frac{\partial f}{\partial y}(x_0, y_0) = \lim_{\Delta y \to 0} \frac{f(x_0, y_0+\Delta y)-f(x_0, y_0)}{\Delta y}$

Using a different notation, $\frac{\partial f}{\partial y}(x, y) = \lim_{h \to 0} \frac{f(x, y+h)-f(x, y)}{h}$

Geometric Interpretation

Geometrically, the partial derivative $\frac{\partial f}{\partial x}(x_0, y_0)$ at a point (x0, y0) can be interpreted as the slope of the tangent line to the curve formed by the intersection of the surface defined by f(x,y) and the plane y = y0. This plane is parallel to the xz-plane and passes through the point (x0, y0).

Similarly, $\frac{\partial f}{\partial y}(x_0, y_0)$ represents the slope of the tangent line to the curve formed by the intersection of the surface defined by f(x,y) and the plane x = x0 This plane is parallel to the yz-plane and passes through the point (x0, y0).(Figure A).

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Solved Examples of Partial Derivatives

Find all the 1st order partial derivatives of the following functions:

Local Extrema and Critical Points in Multivariable Functions

Understanding local minimums and maximums in multivariable functions involves examining the behavior of the function’s partial derivatives.

Local Minimums and Maximums

A function f(x, y) has a relative minimum (maximum) at the point (a, b) if f(x, y) ≥ f(a, b) (f(x, y) ≤ f(a, b)) for all points (x, y) in some region around (a, b). ⚠ Outside of this region, it is completely possible for the function to be smaller (bigger) than f(a, b).

At a local minimum or maximum of a function f(x,y), the partial derivatives with respect to each variable are zero: $\frac{\partial f}{\partial x}(x_0, y_0) = 0,\text{ and } \frac{\partial f}{\partial y}(x_0, y_0) = 0$.

This condition indicates that the tangent plane to the graph z = f(x, y) is horizontal at (x0), y0 ($Δz ≈ f_xΔx+f_yΔy = 0$, Figure C). A horizontal tangent plane means there is no instantaneous rate of change in any direction from this point.

However, it’s essential to understand that this condition is necessary for identifying local extrema (minimums and maximums), it is not sufficient on its own. Critical points can also be saddle points or points of inflection, not just local extrema.

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Critical points in multivariable functions can indeed be local minimums, local maximums or saddle points where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the function, e.g., a relative minimum along one axial direction and a relative maximum along the crossing axis, f(x, y) = x2 - y2, $\frac{\partial f}{\partial x} = 2x, \frac{\partial f}{\partial y} = -2y$ has a critical point at (0, 0) that is a saddle point or point of inflection.

This point is a saddle point because f is a hyperbolic paraboloid. It curves upwards in one direction and downwards in another direction.

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In single-variable calculus, critical points of a function y = f(x) are the x-values within the domain (D) of f where the first derivate is either zero (f’(x) = 0) or undefined. These points can indicate local minima, local maxima, or points of inflection.

Definition. Let f(x1, x2, ···, xn) be a function of n variables. A point (x1, x2, ···, xn) in the domain of f is considered a critical point if all its partial derivatives are zero or if the partial derivatives do not exist. This can be mathematically expressed as: $\frac{∂f}{∂x_1} = 0, \frac{∂f}{∂x_2} = 0, ···, \frac{∂f}{∂x_n} = 0$.

The gradient ∇f is a vector of the partial derivatives of f: ∇f = ($\frac{∂f}{∂x_1}, \frac{∂f}{∂x_2}, ···, \frac{∂f}{∂x_n}$).

At a critical point, the gradient of f at that point is the zero vector or does not exist.

For a function f(x, y) of two variables, a point (x0, y0) is a critical point if $\frac{\partial f}{\partial x}(x_0, y_0) = 0$ and $\frac{\partial f}{\partial y}(x_0, y_0) = 0$ or if these partial derivatives do not exist.

Theorem. If the point (x0, y0) is a local (relative) extremum of the function f(x, y) and the first order derivative of f(x, y) exist at (x0, y0), then (x0, y0) is also a critical point of f and $∇f(x_0, y_0) = \vec{0}$.

Proof.

Consider the new function g(x) = f(x, y0). This is essentially a cross-section of f(x, y) where y is held constant at y0.

Suppose that f(x, y) has a relative (local) extremum at the point (x0, y0). This means that the function g(x) also has a local extremum at x = x0. This is because g(x) is just the function f along the line y = y0.

By Fermat’s Theorem, if a function has a local extremum at a point and its derivative exists at that point, then the derivative at that point must be zero. Therefore, g’(x0) = 0.

The derivative of g(x) with respect to x at x0 is the same as the partial derivative of f with respect to x at (x0, y0): g’(x0) = fx(x0, y0).

Notice that $g’(x) = \frac{∂f}{∂x}(x, y_0)$. When we evaluate this at x = x0, we get $g’(x_0) = \frac{∂f}{∂x}(x_0, y_0)$

Since g’(x0) = 0, it follows that fx(x0, y0) = 0.

Similarly, consider the function h(y) = f(a, y), which is the cross-section of f where x is held constant at x0. By the same reasoning as above, if f(x, y) has a local extremum at (x0, y0), then h(y) has a local extremum at y = y0. By Fermat’s Theorem: h’(y0) = 0.

The derivative of h(y) with respect to y at y0 is the same as the partial derivative of f with respect to y at (x0, y0): h’(y0) = fy(x0, y0).

Since h’(y0) = 0, it follows that fy(x0, y0) = 0.

Since both partial derivatives of f are zero at (x0, y0): $∇f(x_0, y_0) = (\frac{∂f}{∂x}(x_0, y_0), \frac{∂f}{∂y}(x_0, y_0)) = ⟨0, 0⟩ = \vec{0}$. This means that (x0, y0) is a critical point of f.

Bibliography

This content is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License and is based on MIT OpenCourseWare [18.01 Single Variable Calculus, Fall 2007].
  1. NPTEL-NOC IITM, Introduction to Galois Theory.
  2. Algebra, Second Edition, by Michael Artin.
  3. LibreTexts, Calculus. Abstract and Geometric Algebra, Abstract Algebra: Theory and Applications (Judson).
  4. Field and Galois Theory, by Patrick Morandi. Springer.
  5. Michael Penn, Andrew Misseldine, blackpenredpen, and MathMajor, YouTube’s channels.
  6. Contemporary Abstract Algebra, Joseph, A. Gallian.
  7. MIT OpenCourseWare, 18.01 Single Variable Calculus, Fall 2007 and 18.02 Multivariable Calculus, Fall 2007, YouTube.
  8. Calculus Early Transcendentals: Differential & Multi-Variable Calculus for Social Sciences.
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