Advanced Calculus using Mathematica

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

Implicit 2D: Procedure 4.2.1

The Theorem on Smooth Formulas 3.3.1 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_34.png, and AdvCalcMathWeb_35.png are valid in a rectangle about our particular point of tangency, X(x,y).

Procedure  4.2.1: Finding the Implicit Tangent Line

Suppose f[x,y] is smooth in a neighborhood of a particular fixed X=X(x,y). To find the equation of the line tangent to the implicit contour curve AdvCalcMathWeb_36.png, at X=X(x,y), for AdvCalcMathWeb_37.pngconstant, do the following steps:

1.  Compute the general total differential of the equation  AdvCalcMathWeb_38.png

(the coefficients AdvCalcMathWeb_40.png, and AdvCalcMathWeb_41.png are functions of x  and y)

The right hand side is zero since the partial derivatives of a constant AdvCalcMathWeb_42.png are zero.  

2.  Calculate the specific values of the partial derivatives at the particular X(x,y), AdvCalcMathWeb_43.png and AdvCalcMathWeb_44.png, verify that at least one partial derivative is not zero, and write the specific total differential at this point:

m·d x+n·d y=0 or   G • d X =0
(the coefficients m and n are specific numbers)

3. (Optional)  If you want the equation of the tangent line in (x,y)-coordinates, do the following.  Say AdvCalcMathWeb_45.png is your particular point of tangency. The local  (d x,d y)-coordinates are related to (x,y)-coordinates by AdvCalcMathWeb_46.png and AdvCalcMathWeb_47.png, so replace  d x with AdvCalcMathWeb_48.png,  and d y with AdvCalcMathWeb_49.png, obtaining the equation:


The result of Step 2 is an implicit line through the (d x,d y)-origin with perpendicular f[x,y].  You can plot the tangent line using this equation in local coordinates (d x,d y) centered at the particular (x,y).  The equation says, “An unknown vector d X =(d x,d y) (with origin at the point of tangency X(x,y))  lies on the tangent line provided it is perpendicular to the particular gradient G(m,n).”

The result of Step 3 is an implicit line through the point AdvCalcMathWeb_51.png with perpendicular AdvCalcMathWeb_52.png.  This equation says, “An unknown vector X (with origin at (x,y)=(0,0))  lies on the tangent line provided its displacement from AdvCalcMathWeb_53.png is perpendicular to the particular gradient G(m,n).”  Notice that to correctly evaluate this final expression, you need to perform steps (1), (2), and (3) in that order.  

When AdvCalcMathWeb_54.png, the implicit equation does not define a line.  The explicit graph z=f[x,y] has a horizontal tangent plane at a point with zero gradient, but the level set f[x,y]=c may be a curve that does not have a tangent (or other things).  We take this up in the subsection below and in Chapter 5 on Implicit and Inverse Functions.  The rigorous basis for this procedure is in Theorem 5.2.2 below.  It says, if you can compute the tangent line, then the contour is a smooth curve near the point of tangency.


Find the line tangent to the contours of AdvCalcMathWeb_55.png at  (2,1), (-1,2), and (-2,-1).

Solution at X(2,1)

Step 1: The contour AdvCalcMathWeb_56.png has symbolic total differential

AdvCalcMathWeb_57.png  and symbolic gradient AdvCalcMathWeb_58.png

The function and partial derivatives are defined everywhere, so the function is smooth.

Step 2: At (x,y)=(2,1), the specific total differential is

AdvCalcMathWeb_59.png with specific gradient AdvCalcMathWeb_60.png

We sketch this line on the contour plot of the function by drawing the gradient vector with its tail at the point (x,y)=(2,1) and filling in the  perpendicular line as shown next.


Figure 4.4.2: Contour Plot of AdvCalcMathWeb_62.png and the tangent line  at (2,1)