Advanced Calculus using Mathematica

Advanced Calculus using Mathematica: NoteBook Edition is a complete text on calculus of several variables written in Mathematica NoteBooks.

Explicit 3D: Procedure 3.3.1

The Theorem on Smooth Formulas is easy to apply and it tells us that this microscopic view is true.  It boils down to a procedure: compute and check that your computation makes sense.   To prove smoothness when the coordinate function is given by a classical formula, verify that the formulas f[x,y], AdvCalcMathWeb_2.png, and AdvCalcMathWeb_3.png are valid in a rectangle about our particular point of tangency, X(x,y).

Procedure  3.3.1: Finding the Explicit Tangent Plane

Suppose f[x,y] is smooth in a neighborhood of a particular fixed X=X(x,y).  To find the equation of the plane tangent to the explicit surface z=f[x,y] above X(x,y) do the following steps:

1.  Compute the general symbolic total differential


(the partials are functions of  x  and  y)

2.  Calculate the specific values of the partial derivatives at the particular X(x,y), AdvCalcMathWeb_5.png and AdvCalcMathWeb_6.png, and write the specific total differential at this point:

d z=m·d x+n·d y  

(the coefficients m and n are specific numbers)

3. (Optional)  If you want the equation of the tangent line in  (x,y,z)-coordinates, do the following.  Say AdvCalcMathWeb_7.png is your particular point of tangency. The local  (d x,d y,d z)-coordinates are related to (x,y,z)-coordinates by AdvCalcMathWeb_8.png, AdvCalcMathWeb_9.png, AdvCalcMathWeb_10.png so replace  d x with AdvCalcMathWeb_11.png, d y with AdvCalcMathWeb_12.png, and d z wit AdvCalcMathWeb_13.png obtaining the x-y-z-equation of the tangent to the explicit surface z=f[x,y] at AdvCalcMathWeb_14.png


You can plot the tangent plane using the equation from Step 2 in local (d x,d y,d z)-coordinates centered on the surface at the particular (x,y,f[x,y]).  The graph of this equation is an explicit plane through the (d x,d y,d z)-origin centered at the point (x,y,f[x,y]).  The d x-d z-slope is  m  and the  d y-d z-slope is  n.

The result of Step 3 is an explicit (x,y,z)-plane through the fixed point AdvCalcMathWeb_16.png with x-z-slope  AdvCalcMathWeb_17.png  and  y-z-slope  AdvCalcMathWeb_18.png.


Find the equation of the tangent to the explicit graph AdvCalcMathWeb_19.png above (x,y)=(1/3,2) and show that the graph is smooth near this point.


1.  AdvCalcMathWeb_20.png and AdvCalcMathWeb_21.png

These partial derivative functions and the original f[x,y] are defined for all (x,y), so the function is smooth everywhere and has symbolic total differential


2.  Substituting (x,y)=(1/3,2) in the symbolic total differential, we get the particular local equation of the tangent at (1/3,2):


3.  We can replace the local variables if we want the tangent equation in x-y-z-variables,



The smoothness result of step 1 above means the plane approximates the surface “locally” or when magnified as illustrated next.


Figure 3.3.1: Zoomed in (click on graph)

In the eTopic below we explore the local linear approximation by “Zooming” with Mathematica.  In Section 3.4 we will practice differentiation skills with the help of Mathematica.

Mathematica Computation

Define the function <enter>:


Calculate AdvCalcMathWeb_28.png <enter>:


Calculate AdvCalcMathWeb_30.png <enter>:


Print the symbolic differential <enter> (after entering the 3 inputs above):


Specify x and y and print the differential: