\( \newcommand{\sech}{\operatorname{sech}} \) \( \newcommand{\inverse}[1]{#1^\leftarrow} \) \( \newcommand{\<}{\langle} \) \( \newcommand{\>}{\rangle} \) \( \newcommand{\vect}{\mathbf} \) \( \newcommand{\veci}{\mathbf{\hat ı}} \) \( \newcommand{\vecj}{\mathbf{\hat ȷ}} \) \( \newcommand{\veck}{\mathbf{\hat k}} \) \( \newcommand{\curl}{\operatorname{curl}\,} \) \( \newcommand{\dv}{\operatorname{div}\,} \) \( \newcommand{\detThree}[9]{ \operatorname{det}\left( \begin{array}{c c c} #1 & #2 & #3 \\ #4 & #5 & #6 \\ #7 & #8 & #9 \end{array} \right) } \) \( \newcommand{\detTwo}[4]{ \operatorname{det}\left( \begin{array}{c c} #1 & #2 \\ #3 & #4 \end{array} \right) } \)

MA 227 Standard C05

Multivariable Calculus

At the end of the course, each student should be able to…

  • C05: MultivarCalc. Compute and apply the partial derivatives, gradient, and directional derivatives of a multivariable real-valued function.

C05. Multivariable Calculus

  • The limit \(\lim_{P\to P_0} f(P)=L\) states that as points closed to \(P_0\) are plugged into \(f(P)\), the value of \(f(P)\) approaches \(L\).
    • Continuous functions satisfy \(\lim_{P\to P_0} f(P)=f(P_0)\) for all points \(P_0\) in their domain.
  • The partial derivative of a function with respect to a variable describes the rate of change of the function as only that variable changes (so all other variables stay constant).
    • For example, the partial derivative with respect to y is defined by \(\frac{\partial f}{\partial y}=f_y= \lim_{\Delta y\to0}\frac{f(x,y+\Delta y,z)-f(x,y,z)}{\Delta y}\).
    • Since other variables are held constant, computing partial derivatives is the same as computing single-variable derivatives with respect to only the appropriate variable.
  • Second (and higher) order partial derivatives may also be considered.
    • Such derivatives are denoted by \(\frac{\partial^2 f}{\partial y\partial x}=f_{xy}\). In this example, the derivative with respect to \(x\) is taken first, then \(y\).
    • The Mixed Derivative Theorem states that for sufficiently nice functions, \(f_{xy}=f_{yx}\) for all variables \(x,y\).
  • The gradient vector collects all partial derivatives of a multivariable function.
    • For \(f(x,y,z)\), the gradient vector is \(\nabla f=\<f_x,f_y,f_z\>\).
  • The directional derivative describes the rate of change of the function as all variables are changed along a line given by a specified unit vector.
    • The directional derivative with respect to \(\vect u\) is given by \(f_{\vect u}= \lim_{\Delta s\to0}\frac{f(P+\Delta s\vect u)-f(P)}{\Delta s}\).
    • For three variables, this expands to \(f_{\vect u}= \lim_{\Delta s\to0}\frac{f(x+su_x,y+su_y,z+su_z)-f(x,y,z)}{\Delta s}\).
    • For sufficiently nice functions, \(f_{\vect u}=\nabla f \cdot\vect u\).
    • Note, for example, \(f_{\vecj}=f_y\).
    • Thus \(\|\nabla f\|\) is the maximal value of the directional derivative, acheived with \(\vect u=\frac{\nabla f}{\|\nabla f\|}\).
    • Other notations include \(\left(\frac{df}{ds}\right)_{\vect u}\) and \(D_{\vect u}f\).

Textbook References

  • University Calculus: Early Transcendentals (3rd Ed)
    • 13.3 (exercises 1-20, 23-38, 41-54)
    • 13.5 (exercises 1-24)