A function of two variables is just a function whose domain is a subset of and whose range is a subset of i. Another way to visualize the behavior of a function of two variables is to consider its graph. Just as the graph of a function f of one variable is a curve C with equation y = f(x), so the graph of a function f of two variables is a surface S with equation z = f(x, y).
If f is a function of two variables x and y, suppose we allow only x to vary while y remains fixed, for example y = b, where b is a. If g has a derivative at a, we call it the partial derivative of f with respect to x at (a, b) and denote it by fx(a, b). Now if we vary the point (a, b) in equations 2 and 3, fx and fy become functions of two variables.
It is not uncommon in the literature to define a gradient vector as a column vector, following the convention that vectors are generally column vectors. Second, we can immediately apply the multivariate chain rule without paying attention to the dimension of the gradient.
Basic Rules of Partial Differentiation
In the multivariate case where , the basic differentiation rules that we know from school apply (e.g. sum rule, product rule, chain rule; see also section 5.1.2). But when computing derivatives with respect to vectors, we need to be careful: our gradients now involve vectors and matrices, and matrix multiplication is not commutative (Section 2.2.1), i.e. the order matters. The chain rule (5.48) is somewhat similar to the rules for matrix multiplication, where we said that neighboring dimensions must match for matrix multiplication to be defined; see section 2.2.1.
If we multiply the factors together, multiplication is defined, that is, the dimensions of match, and "cancel", such that remains. Note: This is only an intuition, but not mathematically correct, since the partial derivative is not a fraction.
Tangent Planes
Then the tangent plane to the surface S at the point P is defined as the plane containing both tangents. We have seen that if C is any other curve lying on the surface S and passing through P, then its tangent at P also lies in the tangent plane. Therefore you can think that the tangent to S at P consists of all possible tangents.
The tangent plane at P is the plane that most closely approximates the surface S near the point P. We know that any plane passing through the point has an equation of form. The elliptical paraboloid appears to coincide with its tangent plane as we zoom in toward (1, 1, 3).
Notice that the more we zoom in, the flatter the graph appears and the more it resembles its tangent plane. In Figure 3 we confirm this impression by zooming in towards the point (1, 1) on a contour map of the function.
Linear Approximations
In Example 1 we found that an equation of the tangent to the graph of the function is at the point (1, 1, 3). Therefore, in light of the visual evidence in Figures 2 and 3, the linear function of two variables. In general, we know from this that an equation of the tangent to the graph of a function f of two variables is at the point.
Now consider a function of two variables, and suppose that x varies from a to and y varies from b to. In other words, the tangent plane approximates the graph of f well near the point of tangency. It is sometimes difficult to use Definition 7 directly to check the differentiability of a function, but the following theorem provides a convenient sufficient condition for differentiation.
Differentials
Functions of Three or More Variables
Chain Rule
For functions of more than one variable, the Chain Rule has several versions, each of which provides a rule for differentiating a composite function. This means that z is indirectly a function of t, z = f(g(t), h(t)), and the chain rule gives a formula to differentiate z as a. We know that this is the case when fx and fy are continuous according to the following theorem.
Since we often write in place of , we can rewrite the Chain Rule in the form Putting these expressions into Equation 5 and using the equality of the mixed second-order derivatives, we get To calculate the gradient of f with respect to t, we must apply the chain rule (5.48) for multivariate functions as.
This compact way of writing the chain rule as a matrix multiplication only makes sense if the gradient is defined as a row vector.
Gradients of Vector-Valued Functions
We can find this scaling factor by finding a map that transforms the unit square to the other square. In our case, the mapping is linear, and the absolute value of the determinant of this mapping gives us exactly the scaling factor we are looking for. First, we exploit that the mapping is linear so that we can use the tools from Chapter 2.
Second, we will find the mapping using partial derivatives using the tools we discussed in this chapter. It is correct if the coordinate transformation is linear (as in our case), and (5.66) exactly recovers the change-of-basis matrix in (5.62). The absolute value of the Jacobian determinant |det(J)| is the factor by which areas or volumes are scaled when the coordinates are transformed.
In one-dimensional calculus we often use a change of variable (a substitution) to simplify an integral. In general, we consider a change of variables that is given by a transformation T from the UV plane to. We usually assume that T is a C1 transformation, which means that g and h have first-order partial derivatives.
We start with a small rectangle S in the uv plane whose lower left corner is the point (u0, v0) and whose dimensions are and. The image of S is a region R in the xy-plane whose boundary points are Next, we divide a region S in the uv plane into rectangles Sij and call their images in the xy plane Rij.
To find the region S in the UV plane corresponding to R, we note that the sides of R lie on the lines.
