(1) (a) Find the equations of least four different level curves for the surface 2= = f(x,y), f(x, y) = xy. Sketch these level curves on the same graph, so that your illustration starts to look like a topographic map. Label the level curves with the corresponding z-values. (b) A marble is placed on the surface directly above the point (x, y) = (1,−1). (This is the point (1,-1,-1).) Which direction will the marble roll? Describe its direction using a 2-D vector in the xy-plane. Hint: it will roll in the direction of steepest descent. This is a direction orthogonal to the level curve through the point (1,-1).
(1) (a) Find the equations of least four different level curves for the surface 2= = f(x,y), f(x, y) = xy. Sketch these level curves on the same graph, so that your illustration starts to look like a topographic map. Label the level curves with the corresponding z-values. (b) A marble is placed on the surface directly above the point (x, y) = (1,−1). (This is the point (1,-1,-1).) Which direction will the marble roll? Describe its direction using a 2-D vector in the xy-plane. Hint: it will roll in the direction of steepest descent. This is a direction orthogonal to the level curve through the point (1,-1).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![(1)
(a) Find the equations of at least four different level curves for the surface
\[ z = f(x, y), \quad f(x, y) = xy. \]
Sketch these level curves on the same graph, so that your illustration starts to look like a topographic map. Label the level curves with the corresponding \( z \)-values.
(b) A marble is placed on the surface directly above the point \( (x, y) = (1, -1). \) (This is the point \( (1, -1, -1) \).) Which direction will the marble roll? Describe its direction using a 2-D vector in the \( xy \)-plane. Hint: it will roll in the direction of steepest descent. This is a direction orthogonal to the level curve through the point \( (1, -1) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7dc5cf65-67df-4a3e-8bae-308a2a2509c4%2Fa1e6810d-fa76-4528-8385-348c859c38f5%2F57mjiub_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(1)
(a) Find the equations of at least four different level curves for the surface
\[ z = f(x, y), \quad f(x, y) = xy. \]
Sketch these level curves on the same graph, so that your illustration starts to look like a topographic map. Label the level curves with the corresponding \( z \)-values.
(b) A marble is placed on the surface directly above the point \( (x, y) = (1, -1). \) (This is the point \( (1, -1, -1) \).) Which direction will the marble roll? Describe its direction using a 2-D vector in the \( xy \)-plane. Hint: it will roll in the direction of steepest descent. This is a direction orthogonal to the level curve through the point \( (1, -1) \).
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