{"id":1045,"date":"2025-10-15T13:37:08","date_gmt":"2025-10-15T13:37:08","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&p=1045"},"modified":"2025-10-23T13:30:29","modified_gmt":"2025-10-23T13:30:29","slug":"functions-of-several-variables-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/functions-of-several-variables-learn-it-4\/","title":{"raw":"Functions of Several Variables: Learn It 4","rendered":"Functions of Several Variables: Learn It 4"},"content":{"raw":"

Functions of More Than Two Variables<\/h2>\r\n

So far, we have examined only functions of two variables. However, many real-world situations require functions of three or more variables. Consider these examples:<\/p>\r\n

[latex]f(x,y,z)=x^{2}-2xy+y^{2}+3yz-z^{2}+4x-2y+3z-6[\/latex] (a polynomial in three variables)<\/p>\r\n

[latex]g(x,y,t)=(x^{2}-4xy+y^{2})\\sin t-(3x+5y)\\cos t[\/latex]<\/p>\r\n

In the first function, [latex](x,\\ y,\\ z)[\/latex] represents a point in space, and the function [latex]f[\/latex] maps each point to a fourth quantity, such as temperature or wind speed. In the second function, [latex](x,\\ y)[\/latex] represents a point in the plane and [latex]t[\/latex] represents time. This function might map a point in the plane to a third quantity (such as pressure) at a given time [latex]t[\/latex].<\/p>\r\n

The method for finding the domain of a function of more than two variables follows the same principles as for functions of one or two variables.<\/p>\r\n\r\n

Find the domain of each of the following functions:\r\n
    \r\n \t
  1. [latex]f\\,(x,\\ y,\\ z)=\\frac{3x-4y+2z}{\\sqrt{9-x^{2}-y^{2}-z^{2}}}[\/latex]<\/li>\r\n \t
  2. [latex]g\\,(x,\\ y,\\ t)=\\frac{\\sqrt{2t-4}}{x^{2}-y^{2}}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"fs-id2177793933114\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id2177793933114\"]\r\n
      \r\n \t
    1. For the function [latex]f\\,(x,\\ y,\\ z)=\\frac{3x-4y+2z}{\\sqrt{9-x^{2}-y^{2}-z^{2}}}[\/latex] to be defined\u00a0(and be a real value), two conditions must hold:\r\n
        \r\n \t
      1. The denominator cannot be zero<\/li>\r\n \t
      2. The radicand cannot be zero<\/li>\r\n<\/ol>\r\nCombining\u00a0these conditions leads to the inequality\r\n
        [latex]9-x^{2}-y^{2}-z^{2}\\,>\\,0[\/latex].<\/div>\r\n \r\n\r\nMoving the variables\u00a0to the other side and reversing the inequality gives the domain as\r\n
        [latex]\\text{domain}\\ (f)=\\{(x,\\ y,\\ z)\\in\\mathbb{R}^{3}|x^{2}+y^{2}+z^{2}\\,<\\,9\\}[\/latex],<\/div>\r\n \r\n\r\nwhich describes a ball of radius [latex]3[\/latex]\u00a0centered at the origin. (Note<\/em>: The surface of the ball is not included in this domain.)<\/li>\r\n \t
      3. For the function [latex]g\\,(x,\\ y,\\ t)=\\frac{\\sqrt{2t-4}}{x^{2}-y^{2}}[\/latex] to be defined\u00a0(and be a real value), two conditions must hold:\r\n
          \r\n \t
        1. The radicand cannot be negative.<\/li>\r\n \t
        2. The denominator cannot be zero.<\/li>\r\n<\/ol>\r\nSince the radicand cannot be negative, this implies [latex]2t-4\\,\\geq\\,0[\/latex], and therefore that [latex]t\\,\\geq\\,2[\/latex].\u00a0Since the denominator cannot be zero, [latex]x^{2}-y^{2}\\neq{0}[\/latex], or [latex]x^{2}\\neq{y}^{2}[\/latex], which can be rewritten as [latex]y\\neq\\pm{x}[\/latex],\u00a0which are the equations of two lines passing through the origin. Therefore, the domain of [latex]g[\/latex] is\r\n
          [latex]\\text{domain}\\ (g)=\\{(x,\\ y,\\ t)|y\\neq\\pm{x},\\,t\\,\\geq\\,2\\}[\/latex].<\/div><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section>
          \r\n

          Remember the key restrictions when finding domains:<\/p>\r\n\r\n

            \r\n \t
          • Square roots require nonnegative radicands<\/li>\r\n \t
          • Denominators cannot equal zero<\/li>\r\n \t
          • Logarithms require positive arguments<\/li>\r\n<\/ul>\r\n<\/section>\r\n

            Level Surfaces<\/h3>\r\n

            Functions of two variables have level curves, which appear as curves in the [latex]xy[\/latex]-plane. When the function has three variables, these curves extend into three dimensions and become level surfaces<\/strong>.<\/p>\r\n\r\n

            \r\n

            level surfaces<\/h3>\r\nGiven a function [latex]f(x,y,z)[\/latex] and a number [latex]c[\/latex] in the range of [latex]f[\/latex], a level surface<\/strong> of a function of three variables is defined to be the set of points satisfying the equation [latex]f(x,y,z)=c[\/latex].\r\n\r\n<\/section>
            Find the level surface for the function [latex]f\\,(x,\\ y,\\ z)=4x^{2}+9y^{2}-z^{2}[\/latex] corresponding to [latex]c=1[\/latex].[reveal-answer q=\"fs-id2168743953124\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id2168743953124\"]\r\n

            The\u00a0level surface is defined by the equation [latex]4x^{2}+9y^{2}-z^{2}=1[\/latex].\u00a0This equation describes a hyperboloid of one sheet as shown in the following figure.<\/p>\r\n\r\n\r\n[caption id=\"attachment_965\" align=\"aligncenter\" width=\"875\"]\"This Figure 1.\u00a0A hyperboloid of one sheet with some of its level surfaces.[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

            Level surfaces are the three-dimensional analog of level curves. Just as level curves show where a two-variable function has constant value, level surfaces show where a three-variable function has constant value. Different values of [latex]c[\/latex] produce different level surfaces.\r\n\r\n<\/section>","rendered":"

            Functions of More Than Two Variables<\/h2>\n

            So far, we have examined only functions of two variables. However, many real-world situations require functions of three or more variables. Consider these examples:<\/p>\n

            [latex]f(x,y,z)=x^{2}-2xy+y^{2}+3yz-z^{2}+4x-2y+3z-6[\/latex] (a polynomial in three variables)<\/p>\n

            [latex]g(x,y,t)=(x^{2}-4xy+y^{2})\\sin t-(3x+5y)\\cos t[\/latex]<\/p>\n

            In the first function, [latex](x,\\ y,\\ z)[\/latex] represents a point in space, and the function [latex]f[\/latex] maps each point to a fourth quantity, such as temperature or wind speed. In the second function, [latex](x,\\ y)[\/latex] represents a point in the plane and [latex]t[\/latex] represents time. This function might map a point in the plane to a third quantity (such as pressure) at a given time [latex]t[\/latex].<\/p>\n

            The method for finding the domain of a function of more than two variables follows the same principles as for functions of one or two variables.<\/p>\n

            Find the domain of each of the following functions:<\/p>\n
              \n
            1. [latex]f\\,(x,\\ y,\\ z)=\\frac{3x-4y+2z}{\\sqrt{9-x^{2}-y^{2}-z^{2}}}[\/latex]<\/li>\n
            2. [latex]g\\,(x,\\ y,\\ t)=\\frac{\\sqrt{2t-4}}{x^{2}-y^{2}}[\/latex]<\/li>\n<\/ol>\n