Section145GradientsandDirectionalDerivativesinSpace145节梯度和空间定向衍生物

Click to edit Master title style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,*,Section 14.5Gradients and Directional Derivatives in Space,Directional Derivative of Function of 3 Variables,Let,w,=,f,(,x,y,z,)and let the partial derivatives of,f,exist at(,a,b,c,)in its domain.Let be a unit vector.Then:,Now,using a similar argument to last section we can show that,where,is the angle between the gradient and the unit vector,Similar to before we have,If,=0,then our direction is the same as the gradient and,If,=,then our direction is the opposite of the gradient and,If,=,/2,then our direction is the perpendicular to the gradient and,When we had a function of 2 variables,if the directional derivative was zero we were moving in a direction tangent to the level curve and the gradient was perpendicular to the level curve,With a function of 3 variables,if the directional derivative is zero we are moving in a direction tangent to the level surface and the gradient is perpendicular to that level surface,Example,Find,Find the derivative in the direction of,Find the maximum rate of change of,f,at(3,4,5),Find the vector in the direction of the maximum rate of change at(3,4,5),Finding a Tangent Plane,Determine the equation of the plane tangent to at(1,1),Now consider,f,the function of a level surface of where,w,=0,Now we know that the gradient of,w,at a point(,a,b,c,)is perpendicular to,w,at that point,Therefore we have,Now find the tangent plane using the gradient,Lets make sure they are the same,。