# 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)