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)