Calculus
10th Edition
ISBN: 9781285057095
Author: Ron Larson, Bruce H. Edwards
Publisher: Cengage Learning
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Chapter 15.1, Problem 24E
To determine
The conservative vector field for the given potential function h (x, y, z) = x arcsin ( y z )
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- Finding potential functions Determine whether the following vector fields are conservative on the specified region. If so, determine a potential function. Let R* and D* be open regions of ℝ2 and ℝ3, respectively, that do not include the origin.arrow_forwardUsing Green's Theorem on this vector field problem, compute a) the circulation on the boundary of R in terms of a and b, and b) the outward flux across the boundary of R in terms of a and b.arrow_forwardFinding potential functions Determine whether the following vector fields are conservative on the specified region. If so, determine a potential function. Let R* and D* be open regions of ℝ2 and ℝ3, respectively, that do not include the origin. F = ⟨ez, ez, ez (x - y)⟩ on ℝ3arrow_forward
- Splitting a vector field Express the vector field F = ⟨xy, 0, 0⟩in the form V + W, where ∇ ⋅ V = 0 and ∇ x W = 0.arrow_forwarda. Show that the outward flux of the position vector field F = x i + y j + z k through a smooth closed surface S is three times the volume of the region enclosed by the surface. b. Let n be the outward unit normal vector field on S. Show that it is not possible for F to be orthogonal to n at every point of Sarrow_forwardFinding potential functions Determine whether the following vector fields are conservative on the specified region. If so, determine a potential function. Let R* and D* be open regions of ℝ2 and ℝ3, respectively, that do not include the origin. F = ⟨y + z, x + z, x + y⟩ on ℝ3arrow_forward
- Finding potential functions Determine whether the following vector fields are conservative on the specified region. If so, determine a potential function. Let R* and D* be open regions of ℝ2 and ℝ3, respectively, that do not include the origin. F = ⟨z, 1, x⟩ on ℝ3arrow_forwardFinding potential functions Determine whether the following vector fields are conservative on the specified region. If so, determine a potential function. Let R* and D* be open regions of ℝ2 and ℝ3, respectively, that do not include the origin. F = ⟨1, -z, y⟩ on ℝ3arrow_forwardFlux of a vector field? Let S be a closed surface consisting of a paraboloid (z = x²+y²), with (0≤z≤1), and capped by the disc (x²+y² ≤1) on the plane (z=1). Determine the flow of the vector field F (x,y,z) = zj − yk, in the direction that points out across the surface S.arrow_forward
- Finding potential functions Determine whether the following vector fields are conservative on the specified region. If so, determine a potential function. Let R* and D* be open regions of ℝ2 and ℝ3, respectively, that do not include the origin. F = ⟨y, x, x - y⟩ on ℝ3arrow_forwardUsing Gauss' theorem to calculate the flow of the vector field 3x3 F: F (x, y, z) = (x^2z, 2x^2, 3z^2) exiting the cylinder defined from the relations x ^2+y ^2<=1, 1<= z <= 2.arrow_forwarduse Green’s Theorem to find the counterclock-wise circulation and outward flux for the field F and curve C. F = (y2 - x2 )i + (x2 + y2 )j C: The triangle bounded by y = 0, x = 3, and y =x.arrow_forward
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