Tangency Condition and Gradient Vectors in Multivariable Functions

In multivariable calculus, the tangency condition between level surfaces of functions and the relationship between their gradient vectors play a crucial role in understanding the behavior of functions in higher dimensions. Specifically, if the level surfaces of two functions are tangent at a point, then the gradients of these functions at that point are related in a particular way. This article aims to explore this relationship and provide a comprehensive explanation through both intuitive reasoning and formal mathematics.

Understanding Level Surfaces and Tangency

Consider two functions, f(x, y, z) c and g(x, y, z) d, which represent level surfaces. The level surface of a function is the set of points where the function takes a constant value. Let's focus on the case where these level surfaces are tangent at a point (x_0, y_0, z_0). This tangency condition implies that the surfaces share a common tangent plane at this point.

Intuitive Reasoning: Tangent Planes and Gradients

In the intuitive reasoning, we start by considering two curves in the xy plane given by the level surfaces n f(x, y) c and n g(x, y) d. These curves meet at the point (x_0, y_0) and share the same tangent line at this point.

Let's begin with the idea that along each curve, the value of the function is constant. Therefore, the differential of the function along each curve must be zero:

df f'_x dx f'_y dy 0

dg g'_x dx g'_y dy 0

for any values of dx and dy. This implies that the ratios of the partial derivatives are equal, leading us to:

where p is a common factor. This can be rewritten as:

df p(g'_y dx - g'_x dy)

and

This shows that the gradients of f and g at the point (x_0, y_0) are proportional, i.e.,

If this is extended to x, y, z, we can write:

Formal Reasoning: Implicit Functions and Local Derivatives

Another way to approach this is through the local derivatives of the implicit functions y[f] y_f(x, y) and y[g] y_g(x, y). Since these functions have the same tangent line at the point (x_0, y_0, z_0), their local derivatives (gradients) are equal:

Substituting this into the differential equations, we get:

for any dx. This implies that:

From these, we can deduce that:

Therefore, we can write:

This implies:

Conclusion

The tangency condition between the level surfaces of two functions leads to a direct relationship between their gradients at the point of tangency. This relationship is encapsulated in the equation grad f_0 b grad g_0 0, where a and b are arbitrary constants. Understanding and applying this concept is crucial in multivariable calculus and has wide-ranging applications in various fields of science and engineering.