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The partial derivatives, fx and fy, represent the slopes of tangent lines to a surface in the x-direction and the y-direction. Given the the surface determined by z = 4 - x2 - y2 and the point ( 1, 1, 2 ), fx( 1, 1, 2 ) = -2 and fy( 1, 1, 2 ) = -2. Think of the plane perpendicular to the xy plane parallel to the x-axis and the plane perpendicular to the xy plane parallel to the y-axis each containing the point ( 1, 1, 2 ). The intersections are parabolas containing the point ( 1, 1, 2 ). The slopes of the tangent lines, one in the x direction and the other in the y direction, are fx and fy, respectively.
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Suppose that we wish to find the slope of the tangent line to the surface in any direction. That slope is the directional derivative of f( x, y ) in the direction of a unit vector u = < cos( q ), sin( q ) >. It is denoted by Duf( x, y ) and is equal to fx( x, y )cos( q ) + fy( x, y )sin( q ) = < fx( x, y ), fy( x, y ) > . < cos( q ), sin( q ) >. The first vector in the dot product, < fx( x, y ), fy( x, y ) >, is the gradient of function f( x, y ) and is denoted by &emdash;f, read as del f. The gradient is a two dimensional vector ( in the xy plane ), that points in the direction of maximum change of the function. Its magnitude is the greatest of all possible directional derivatives. The direction of minimum is in the opposite direction. Another important property of the gradient &emdash;f is the fact that it is orthogonal to the level curves at any particular point. Below is the level curve where z = 4 of the function f( x, y ) = x2 + y2, a circle of radius 2. The gradient &emdash;f = < 2x, 2y >. Graphing each gradient vector at points around the level curve illustrates the fact that the gradient is orthogonal to the level curve. If the function is one of three variables then the gradient is a three dimensional vector that points in the direction of maximal change. Each gradient is orthogonal to the level surface at any point.
Let's do another function, f( x, y ) = x2 - y2, a point on the graph, ( -1, 1, 0 ), and a direction in the xy plane, < 2, 1 >.
The plane shown above, x - 2y = -3, contains the point ( -1, 1, 0 ) and is parallel to < 2, 1 >. The directional derivative at ( -1, 1, 0 ) in the direction of < 2, 1 > is the slope of the tangent line to the curve of intersection. That slope is -6/÷5. Check it out! The gradient for the function is < 2x, -2y >. At the point ( -1, 1, 0 ), the gradient is < -2, -2 >. This vector points in the direction of maximal change. The directional derivative in this direction is 2÷2. It is the greatest of all directional derivatives. The least is the directional derivative in the opposite direction. The level curve is the intersection of the function with z = 0, the xy plane. This gives level curve of y = ± x, two intersecting lines. Plotting the gradient at various points along the gradient yields:
