# geometric interpretation of second order partial derivatives

The partial derivative of a function of $$n$$ variables, is itself a function of $$n$$ variables. As we saw in Activity 10.2.5 , the wind chill $$w(v,T)\text{,}$$ in degrees Fahrenheit, is … Michael Hardy. Section 3 Second-order Partial Derivatives. Activity 10.3.4 . Just as with the first-order partial derivatives, we can approximate second-order partial derivatives in the situation where we have only partial information about the function. Do they offer anything "meaningful" in the same way that the first- and unmixed second-order partial derivatives do? Note that a function of three variables does not have a graph. By taking the partial derivatives of the partial derivatives, we compute the higher-order derivatives.Higher-order derivatives are important to check the concavity of a function, to confirm whether an extreme point of a function is max or min, etc. Suppose is a function of two variables which we denote and .There are two possible second-order mixed partial derivative functions for , namely and .In most ordinary situations, these are equal by Clairaut's theorem on equality of mixed partials.Technically, however, they are defined somewhat differently. The partial derivative of a function (,, … Second derivative usually indicates a geometric property called concavity. For the mixed partial, derivative in the x and then y direction (or vice versa by Clairaut's Theorem), would that be the slope in a diagonal direction? share | cite | improve this question | follow | edited Aug 13 '15 at 3:25. And then to get the concavity in the x … the matrix consisting of the second order partial derivatives: $$H(x,y) := \begin{pmatrix} f_{xx} & f_{xy} \\ f_{xy} & f_{yy} \end{pmatrix}. The object that truly has geometric meaning is the Hessian, i.e. Afterwards, the instructor reviews the correct answers with the students in order to correct any misunderstandings concerning the process of finding partial derivatives. Write \mathbf x = (x, y). Geometric interpretation: Partial derivatives of functions of two variables ad-mit a similar geometrical interpretation as for functions of one variable. Purpose The purpose of this lab is to acquaint you with using Maple to compute partial derivatives. Background For a function of a single real variable, the derivative gives information on whether the graph of is increasing or decreasing. The second order partials in the x and y direction would give the concavity of the surface. Definition For a function of two variables. For example, @w=@x means diﬁerentiate with respect to x holding both y and z constant and so, for this example, @w=@x = sin(y + 3z).$$ (In the following, I will denote the dot/scalar product by $\langle(u_1, u_2), (v_1, v_2)\rangle = u_1 v_1 + u_2 v_2$.). calculus partial-derivative geometric-interpretation. In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary).Partial derivatives are used in vector calculus and differential geometry.. Partial derivatives are computed similarly to the two variable case. Partial Derivatives and their Geometric Interpretation.

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