{"id":206,"date":"2021-07-30T17:32:20","date_gmt":"2021-07-30T17:32:20","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&p=206"},"modified":"2022-11-01T05:38:15","modified_gmt":"2022-11-01T05:38:15","slug":"putting-it-together-vector-calculus","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/putting-it-together-vector-calculus\/","title":{"raw":"Putting It Together: Vector Calculus","rendered":"Putting It Together: Vector Calculus"},"content":{"raw":"

Drawing a Rotational Vector Field<\/h2>\r\n[caption id=\"attachment_235\" align=\"aligncenter\" width=\"450\"]\"A Figure 1.[\/caption]\r\n\r\nSketch the vector field [latex]{\\bf{F}}(x,y)=\\langle y,-x\\rangle[\/latex]\r\n

Solution<\/h3>\r\nCreate a table (see the one that follows) using a representative sample of points in a plane and their corresponding vectors.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
[latex](x,y)[\/latex]<\/td>\r\n[latex]{\\bf{F}}(x,y)[\/latex]<\/td>\r\n[latex](x,y)[\/latex]<\/td>\r\n[latex]{\\bf{F}}(x,y)[\/latex]<\/td>\r\n[latex](x,y)[\/latex]<\/td>\r\n[latex]{\\bf{F}}(x,y)[\/latex]<\/td>\r\n<\/tr>\r\n
[latex](1,0)[\/latex]<\/td>\r\n[latex]\\langle 0,-1\\rangle[\/latex]<\/td>\r\n[latex](2,0)[\/latex]<\/td>\r\n[latex]\\langle 0,-2 \\rangle[\/latex]<\/td>\r\n[latex](1,1)[\/latex]<\/td>\r\n[latex]\\langle 1,-1\\rangle[\/latex]<\/td>\r\n<\/tr>\r\n
[latex](0,1)[\/latex]<\/td>\r\n[latex]\\langle 1,0\\rangle[\/latex]<\/td>\r\n[latex](0,2)[\/latex]<\/td>\r\n[latex]\\langle 2,0\\rangle[\/latex]<\/td>\r\n[latex](-1,1)[\/latex]<\/td>\r\n[latex]\\langle 1,1\\rangle[\/latex]<\/td>\r\n<\/tr>\r\n
[latex](-1,0)[\/latex]<\/td>\r\n[latex]\\langle 0,1\\rangle[\/latex]<\/td>\r\n[latex](-2,0)[\/latex]<\/td>\r\n[latex]\\langle 0,2\\rangle[\/latex]<\/td>\r\n[latex](-1,-1)[\/latex]<\/td>\r\n[latex]\\langle -1,1\\rangle[\/latex]<\/td>\r\n<\/tr>\r\n
[latex](0,-1)[\/latex]<\/td>\r\n[latex]\\langle -1,0\\rangle[\/latex]<\/td>\r\n[latex](0,-2)[\/latex]<\/td>\r\n[latex]\\langle -2,0\\rangle[\/latex]<\/td>\r\n[latex](1,-1)[\/latex]<\/td>\r\n\u00a0[latex]\\langle -1,-1\\rangle[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[caption id=\"attachment_331\" align=\"aligncenter\" width=\"740\"]\"A Figure 2. (a) A visual representation of vector field [latex]{\\bf{F}}(x,y)=\\langle y,-x\\rangle[\/latex]. (b) Vector field [latex]{\\bf{F}}(x,y)=\\langle y,-x\\rangle[\/latex] with circles centered at the origin. (c) Vector [latex]{\\bf{F}}(a,b)[\/latex] is perpendicular to radial vector [latex]\\langle a,b\\rangle[\/latex] at point [latex](a,b)[\/latex].[\/caption]\r\n

Analysis<\/h3>\r\nNote that vector [latex]{\\bf{F}}(a,b)=\\langle b,-a\\rangle[\/latex]\u00a0points clockwise and is perpendicular to radial vector [latex]\\langle a,b\\rangle[\/latex].\u00a0(We can verify this assertion by computing the dot product of the two vectors: [latex]\\langle a,b\\rangle\\cdot\\langle -b,a\\rangle= -ab+ab=0[\/latex].) Furthermore, vector [latex]\\langle b,-a\\rangle[\/latex] has length [latex]r=\\sqrt{a^{2}+b^{2}}[\/latex]. Thus, we have a complete description of this rotational vector field: the vector associated with point [latex](a,b)[\/latex] is the vector with length [latex]r[\/latex]\u00a0tangent to the circle with radius\u00a0[latex]r[\/latex],\u00a0and it points in the clockwise direction.\r\n\r\nSketches such as that in\u00a0Figure 6 under Example \"Sketching a Vector Field\"\u00a0are often used to analyze major storm systems, including\u00a0hurricanes<\/span>\u00a0and cyclones. In the northern hemisphere, storms rotate counterclockwise; in the southern hemisphere, storms rotate clockwise. (This is an effect caused by Earth\u2019s rotation about its axis and is called the Coriolis Effect.)","rendered":"

Drawing a Rotational Vector Field<\/h2>\n
\"A<\/p>\n

Figure 1.<\/p>\n<\/div>\n

Sketch the vector field [latex]{\\bf{F}}(x,y)=\\langle y,-x\\rangle[\/latex]<\/p>\n

Solution<\/h3>\n

Create a table (see the one that follows) using a representative sample of points in a plane and their corresponding vectors.<\/p>\n\n\n\n\n\n\n\n
[latex](x,y)[\/latex]<\/td>\n[latex]{\\bf{F}}(x,y)[\/latex]<\/td>\n[latex](x,y)[\/latex]<\/td>\n[latex]{\\bf{F}}(x,y)[\/latex]<\/td>\n[latex](x,y)[\/latex]<\/td>\n[latex]{\\bf{F}}(x,y)[\/latex]<\/td>\n<\/tr>\n
[latex](1,0)[\/latex]<\/td>\n[latex]\\langle 0,-1\\rangle[\/latex]<\/td>\n[latex](2,0)[\/latex]<\/td>\n[latex]\\langle 0,-2 \\rangle[\/latex]<\/td>\n[latex](1,1)[\/latex]<\/td>\n[latex]\\langle 1,-1\\rangle[\/latex]<\/td>\n<\/tr>\n
[latex](0,1)[\/latex]<\/td>\n[latex]\\langle 1,0\\rangle[\/latex]<\/td>\n[latex](0,2)[\/latex]<\/td>\n[latex]\\langle 2,0\\rangle[\/latex]<\/td>\n[latex](-1,1)[\/latex]<\/td>\n[latex]\\langle 1,1\\rangle[\/latex]<\/td>\n<\/tr>\n
[latex](-1,0)[\/latex]<\/td>\n[latex]\\langle 0,1\\rangle[\/latex]<\/td>\n[latex](-2,0)[\/latex]<\/td>\n[latex]\\langle 0,2\\rangle[\/latex]<\/td>\n[latex](-1,-1)[\/latex]<\/td>\n[latex]\\langle -1,1\\rangle[\/latex]<\/td>\n<\/tr>\n
[latex](0,-1)[\/latex]<\/td>\n[latex]\\langle -1,0\\rangle[\/latex]<\/td>\n[latex](0,-2)[\/latex]<\/td>\n[latex]\\langle -2,0\\rangle[\/latex]<\/td>\n[latex](1,-1)[\/latex]<\/td>\n\u00a0[latex]\\langle -1,-1\\rangle[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
\"A<\/p>\n

Figure 2. (a) A visual representation of vector field [latex]{\\bf{F}}(x,y)=\\langle y,-x\\rangle[\/latex]. (b) Vector field [latex]{\\bf{F}}(x,y)=\\langle y,-x\\rangle[\/latex] with circles centered at the origin. (c) Vector [latex]{\\bf{F}}(a,b)[\/latex] is perpendicular to radial vector [latex]\\langle a,b\\rangle[\/latex] at point [latex](a,b)[\/latex].<\/p>\n<\/div>\n

Analysis<\/h3>\n

Note that vector [latex]{\\bf{F}}(a,b)=\\langle b,-a\\rangle[\/latex]\u00a0points clockwise and is perpendicular to radial vector [latex]\\langle a,b\\rangle[\/latex].\u00a0(We can verify this assertion by computing the dot product of the two vectors: [latex]\\langle a,b\\rangle\\cdot\\langle -b,a\\rangle= -ab+ab=0[\/latex].) Furthermore, vector [latex]\\langle b,-a\\rangle[\/latex] has length [latex]r=\\sqrt{a^{2}+b^{2}}[\/latex]. Thus, we have a complete description of this rotational vector field: the vector associated with point [latex](a,b)[\/latex] is the vector with length [latex]r[\/latex]\u00a0tangent to the circle with radius\u00a0[latex]r[\/latex],\u00a0and it points in the clockwise direction.<\/p>\n

Sketches such as that in\u00a0Figure 6 under Example “Sketching a Vector Field”\u00a0are often used to analyze major storm systems, including\u00a0hurricanes<\/span>\u00a0and cyclones. In the northern hemisphere, storms rotate counterclockwise; in the southern hemisphere, storms rotate clockwise. (This is an effect caused by Earth\u2019s rotation about its axis and is called the Coriolis Effect.)<\/p>\n\n\t\t\t

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