![]() ![]() Want to cite, share, or modify this book? This book uses theĬreative Commons Attribution-NonCommercial-ShareAlike License This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission. f x ( x, y, z ) = −2 e −2 z cos 2 x cos 2 y, f y ( x, y, z ) = −2 e −2 z sin 2 x sin 2 y and f z ( x, y, z ) = −2 e −2 z sin 2 x cos 2 y, so ∇ f ( x, y, z ) = f x ( x, y, z ) i + f y ( x, y, z ) j + f z ( x, y, z ) k = ( 2 e −2 z cos 2 x cos 2 y ) i + ( - 2 e - 2 z sin 2 x sin 2 y ) j + ( 2 e - 2 z sin 2 x cos 2 y ) k = 2 e −2 z ( cos 2 x cos 2 y i − sin 2 x sin 2 y j − sin 2 x cos 2 y k ). į x ( x, y, z ) = −2 e −2 z cos 2 x cos 2 y, f y ( x, y, z ) = −2 e −2 z sin 2 x sin 2 y and f z ( x, y, z ) = −2 e −2 z sin 2 x cos 2 y, so ∇ f ( x, y, z ) = f x ( x, y, z ) i + f y ( x, y, z ) j + f z ( x, y, z ) k = ( 2 e −2 z cos 2 x cos 2 y ) i + ( - 2 e - 2 z sin 2 x sin 2 y ) j + ( 2 e - 2 z sin 2 x cos 2 y ) k = 2 e −2 z ( cos 2 x cos 2 y i − sin 2 x sin 2 y j − sin 2 x cos 2 y k ). and b., we first calculate the partial derivatives f x, f y, f x, f y, and f z, f z, then use Equation 4.40.į x ( x, y, z ) = 10 x − 2 y + 3 z, f y ( x, y, z ) = −2 x + 2 y − 4 z and f z ( x, y, z ) = 3 x − 4 y + 2 z, so ∇ f ( x, y, z ) = f x ( x, y, z ) i + f y ( x, y, z ) j + f z ( x, y, z ) k = ( 10 x − 2 y + 3 z ) i + ( −2 x + 2 y − 4 z ) j + ( 3 x - 4 y + 2 z ) k. These three cases are outlined in the following theorem.įor both parts a. In the first case, the value of D u f ( x 0, y 0 ) D u f ( x 0, y 0 ) is maximized in the second case, the value of D u f ( x 0, y 0 ) D u f ( x 0, y 0 ) is minimized. If φ = π, φ = π, then cos φ = −1 cos φ = −1 and ∇ f ( x 0, y 0 ) ∇ f ( x 0, y 0 ) and u u point in opposite directions. If φ = 0, φ = 0, then cos φ = 1 cos φ = 1 and ∇ f ( x 0, y 0 ) ∇ f ( x 0, y 0 ) and u u both point in the same direction. Recall that cos φ cos φ ranges from −1 −1 to 1. Therefore, the directional derivative is equal to the magnitude of the gradient evaluated at ( x 0, y 0 ) ( x 0, y 0 ) multiplied by cos φ. The ‖ u ‖ ‖ u ‖ disappears because u u is a unit vector. Therefore, the z-coordinate of the second point on the graph is given by z = f ( a + h cos θ, b + h sin θ ). The distance we travel is h h and the direction we travel is given by the unit vector u = ( cos θ ) i + ( sin θ ) j. ![]() We measure the direction using an angle θ, θ, which is measured counterclockwise in the x, y-plane, starting at zero from the positive x-axis ( Figure 4.39). Given a point ( a, b ) ( a, b ) in the domain of f, f, we choose a direction to travel from that point. We start with the graph of a surface defined by the equation z = f ( x, y ). Now we consider the possibility of a tangent line parallel to neither axis. Similarly, ∂ z / ∂ y ∂ z / ∂ y represents the slope of the tangent line parallel to the y -axis. For example, ∂ z / ∂ x ∂ z / ∂ x represents the slope of a tangent line passing through a given point on the surface defined by z = f ( x, y ), z = f ( x, y ), assuming the tangent line is parallel to the x-axis. These derivatives correspond to each of the independent variables and can be interpreted as instantaneous rates of change (that is, as slopes of a tangent line). ![]() A function z = f ( x, y ) z = f ( x, y ) has two partial derivatives: ∂ z / ∂ x ∂ z / ∂ x and ∂ z / ∂ y. ![]() In Partial Derivatives we introduced the partial derivative. 4.6.5 Calculate directional derivatives and gradients in three dimensions.4.6.4 Use the gradient to find the tangent to a level curve of a given function.4.6.3 Explain the significance of the gradient vector with regard to direction of change along a surface.4.6.2 Determine the gradient vector of a given real-valued function.4.6.1 Determine the directional derivative in a given direction for a function of two variables. ![]()
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