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The vector field F(r) is given by F = X Y ¸1 + x² + y² ¹ 1 + x² + y²¹ (a) Sketch the field F in the (x, y)-plane. (b) Showing your workings, explicitly evaluate the line integral fo 2z F.dr where is a circle of radius 2 in the (x, y)-plane, traversed anti-clockwise. (c) Evaluate curl F and state why your answer is consistent with the result of part (b).

The vector field F(r) is given by F = X Y ¸1 + x² + y² ¹ 1 + x² + y²¹ (a) Sketch the field F in the (x, y)-plane. (b) Showing your workings, explicitly evaluate the line integral fo 2z F.dr where is a circle of radius 2 in the (x, y)-plane, traversed anti-clockwise. (c) Evaluate curl F and state why your answer is consistent with the result of part (b).

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter8: Applications Of Trigonometry
Section8.4: The Dot Product
Problem 12E
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2. The vector field F(r) is given by = (₁ F = X Y 1 + x² + y²¹ 1 + x² + y²¹ (a) Sketch the field F in the (x, y)-plane. (b) Showing your workings, explicitly evaluate the line integral fo C d dt F.dr where is a circle of radius 2 in the (x, y)-plane, traversed anti-clockwise. (c) Evaluate curl F and state why your answer is consistent with the result of part (b). (d) Find the scalar potential of F and sketch some contours of in the (x, y)-plane. You may find it useful to recall that the derivative of the natural logarithm of a function h(t) is given by log h J = = 2z 1 dh h dt (e) Calculate the divergence of F. Hence, without explicitly evaluating a surface integral, state whether J is positive, negative or zero, where J is defined as the integral $ S F. dS over the surface of a sphere of radius 3, centred on the point (1,1,1). Explain your reasoning by referring to a relevant theorem.
Transcribed Image Text:2. The vector field F(r) is given by = (₁ F = X Y 1 + x² + y²¹ 1 + x² + y²¹ (a) Sketch the field F in the (x, y)-plane. (b) Showing your workings, explicitly evaluate the line integral fo C d dt F.dr where is a circle of radius 2 in the (x, y)-plane, traversed anti-clockwise. (c) Evaluate curl F and state why your answer is consistent with the result of part (b). (d) Find the scalar potential of F and sketch some contours of in the (x, y)-plane. You may find it useful to recall that the derivative of the natural logarithm of a function h(t) is given by log h J = = 2z 1 dh h dt (e) Calculate the divergence of F. Hence, without explicitly evaluating a surface integral, state whether J is positive, negative or zero, where J is defined as the integral $ S F. dS over the surface of a sphere of radius 3, centred on the point (1,1,1). Explain your reasoning by referring to a relevant theorem.
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