Parametric Equations
Parametric equations express a set of related quantities as explicit functions of an independent parameter. In multivariable calculus, these equations are especially useful for defining curves and surfaces in three-dimensional space. A simple example would be \(x(t) = a + t, y(t) = b + t\), which represents a line in two dimensions, where \(t\) is the parameter.
When examining the function given in the exercise, \(f(x, y)=\frac{4 x y}{(x^{2}+1)(y^{2}+1)}\), we create parametric equations for the normal line to the surface at a specific point. This involves determining a direction vector for the line, which in this case is parallel to the gradient vector at the given point, and expressing the x, y, and z coordinates in terms of a new parameter, say \(t\), based on this direction. The advantage of using parametric equations is that it allows us to trace the path of the curve (or line) in space as \(t\) varies, thus explaining the motion along the curve.
Tangent Plane
The tangent plane to a surface at a given point is the plane that best approximates the surface near that point. For a smooth surface defined by a function \(z = f(x, y)\), the equation of the tangent plane at point \( (x_0, y_0, f(x_0, y_0)) \) can be found using the formula \( f(x_{0}, y_{0}) + f_x(x_{0}, y_{0})(x-x_{0}) + f_y(x_{0}, y_{0})(y-y_{0}) = z\), where \(f_x\) and \(f_y\) are the partial derivatives of \(f\) with respect to \(x\) and \(y\), respectively.
In our exercise, for the point \( (1,1,1) \), the tangent plane is horizontal, indicated by the zero gradients (flat surface). In contrast, at \( (-1,2,-4/5) \), the tangent plane equations incorporate the non-zero gradients, creating a plane sloped in space, reflecting the surface's local inclination around the point.
Normal Line
The normal line at a point on a surface is a line that is perpendicular to the tangent plane at that point. For the function \(z = f(x, y)\), the direction of the normal line can be found using the gradient of \(f\) at \( (x_0, y_0) \), and is often expressed in the parametric form \( (x, y, z) = (x_{0}, y_{0}, z_{0}) + t(abla f(x_{0}, y_{0}))\) where \(t\) is again a parameter and \(z_{0} = f(x_{0}, y_{0})\).
Returning to the textbook problem, at the point \( (1,1,1) \), the normal line equation signifies that the line is vertical, as deduced from the horizontal tangent plane at that point. On the other hand, the normal at \( (-1,2,-4/5) \) calculated from the gradient reflects the direction in which the surface is locally increasing or decreasing most rapidly.
Partial Derivatives
Partial derivatives are an extension of the concept of a derivative to functions of multiple variables. They represent the rate at which a function changes as one specific variable changes, holding all other variables constant. Symbolically, the partial derivative of \( f(x, y) \) with respect to \( x \) is denoted by \( f_x(x, y) \) and similarly \( f_y(x, y) \) for the variable \( y \).
The computation of partial derivatives is a pivotal step in finding the equations for both the tangent planes and the normal lines to surfaces, as seen in the exercise. These derivatives reveal how the function \( f(x, y) \) behaves in terms of changes in \( x \) and \( y \) individually, thus dictating the slope of the tangent plane in each direction at a given point.
Graphing Multivariable Functions
Graphing multivariable functions, such as \(f(x, y)\), is an important visual tool for understanding the behavior of these functions over a particular domain. In the step-by-step solution, the use of a computer algebra system is advised to visualize the surface defined by the function along with its normal lines and tangent planes at specified points.
Graphs help interpret the analytical results for the points \( (1,1,1) \) and \( (-1,2,-4/5) \) by providing a way to see the local geometry of the surface. For instance, we can observe a peak at one point and a local minimum at the other. These graphical insights complement the analytical findings and contribute to a thorough understanding of the surface's properties.
