2.1 Multivariable Functions and Partial Derivatives

Multivariable functions depend on multiple variables, like temperature and pressure. They're often visualized using 3D coordinate systems or contour plots, helping us understand complex relationships in physics and engineering.

Partial derivatives measure how a function changes with respect to one variable while others stay constant. They're crucial for calculating tangent planes, normal lines, and applying the chain rule in multiple dimensions, essential tools in mathematical physics.

Multivariable Functions

Concept of multivariable functions

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• Multivariable functions depend on two or more independent variables (temperature, pressure)
  • Denoted as $f(x_1, x_2, ..., x_n)$ where $n$ represents the number of independent variables
• Graphical representations visualize the relationship between independent and dependent variables
  • 3D coordinate system used for functions with two independent variables ($x$, $y$)
    • $x$ and $y$ axes represent the independent variables (latitude, longitude)
    • $z$ axis represents the dependent variable or function value (elevation)
  • Hypersurfaces represent functions with more than two independent variables
    • Difficult to visualize graphically due to higher dimensions
    • Often represented by level curves or contour plots (topographic maps)

Calculation of partial derivatives

• Partial derivatives measure the rate of change of a multivariable function with respect to one independent variable while treating other variables as constants
  • Denoted as $\frac{\partial f}{\partial x_i}$ where $x_i$ is the variable with respect to which the derivative is taken
• Calculating partial derivatives involves applying standard differentiation rules to the variable of interest while treating other variables as constants
  • Given $f(x, y) = x^2y + \sin(xy)$
    • $\frac{\partial f}{\partial x} = 2xy + y\cos(xy)$ (differentiate with respect to $x$, treat $y$ as constant)
    • $\frac{\partial f}{\partial y} = x^2 + x\cos(xy)$ (differentiate with respect to $y$, treat $x$ as constant)

Applications of Partial Derivatives

Applications of partial derivatives

• Tangent planes are flat surfaces that touch a curved surface at a single point and are parallel to the surface at that point
  • Equation of the tangent plane at $(x_0, y_0, f(x_0, y_0))$: $z - f(x_0, y_0) = \frac{\partial f}{\partial x}(x_0, y_0)(x - x_0) + \frac{\partial f}{\partial y}(x_0, y_0)(y - y_0)$
    • Partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ evaluated at the point of interest $(x_0, y_0)$ determine the slope of the tangent plane
• Normal lines are perpendicular to the tangent plane at a given point
  • Direction vector of the normal line: $\vec{n} = \left(\frac{\partial f}{\partial x}(x_0, y_0), \frac{\partial f}{\partial y}(x_0, y_0), -1\right)$
    • Partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ evaluated at the point $(x_0, y_0)$ determine the direction of the normal line

Chain rule in multiple dimensions

• Chain rule for partial derivatives allows calculation of the partial derivative of a composite function
  • For a composite function $f(x, y) = g(u(x, y), v(x, y))$, the chain rule states:
    1. $\frac{\partial f}{\partial x} = \frac{\partial g}{\partial u}\frac{\partial u}{\partial x} + \frac{\partial g}{\partial v}\frac{\partial v}{\partial x}$
    2. $\frac{\partial f}{\partial y} = \frac{\partial g}{\partial u}\frac{\partial u}{\partial y} + \frac{\partial g}{\partial v}\frac{\partial v}{\partial y}$
• Applying the chain rule involves:
  1. Identifying the outer function $g$ and inner functions $u$ and $v$
  2. Calculating the partial derivatives of $g$ with respect to $u$ and $v$
  3. Calculating the partial derivatives of $u$ and $v$ with respect to $x$ and $y$
  4. Multiplying and adding the appropriate partial derivatives according to the chain rule formulas