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Equation of a tangent plane

  1. Feb 20, 2015 #1
    1. The problem statement, all variables and given/known data
    What is the difference between the two given equations below? When would you use one or the either?

    2. Relevant equations

    ## z - z_1 = \frac{∂z}{∂x}(x_1, y_1, z_1)(x - x_1) + \frac{∂z}{∂y}(x_1, y_1, z_1)(y - y_1) ##
    ## \frac{∂f}{∂x}(x_0, y_0, z_0)(x - x_0) + \frac{∂f}{∂y}(x_0, y_0, z_0)(y - y_0) + \frac{∂f}{∂z}(x_0, y_0, z_0)(z - z_o) ##

    3. The attempt at a solution

    I am confused because it seems like there are several equations for the tangent plane.
  2. jcsd
  3. Feb 20, 2015 #2


    Staff: Mentor

    The two equations above are working with different forms for the equation of the surface on which you're asked to find a tangent plane.

    In the first equation, you are given z as a function of x and y (i.e., z = g(x, y)).
    In the second equation, the surface is given like this: f(x, y, z) = C.
  4. Feb 20, 2015 #3
    thank you
  5. Feb 20, 2015 #4


    User Avatar
    Science Advisor

    You understand, I hope, that the second one is not an equation! I presume that was a typo and it was supposed to be equal to 0 or some other constant.

    Given that, the first is generally used where you are given z as a function of x and y: z= z(x,y), with [itex]z_1= z(x_1, y_1)[/itex].

    The second would be used when you given some function of x, y, and z equal to a constant: [tex]F(x, y, z)= C[/tex].

    That second form is more general since, if you have z= f(x, y), you can write it as F(x, y, z)= z- f(x, y)= 0 but if you are given F(x, y z)= C, you may not be able to solve for z. For example, [tex]F(x, y, z)= x^2+ y^2+ z^2= R^2[/itex] is the equation of a sphere. If you wanted to solve for z, you have to do two cases- [itex]z= \sqrt{R^2- x^2- y^2}['itex] and [itex]z= -\sqrt{R^2- x^2- y^2}[/itex].
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