Jacobian determinant

Examples of Jacobian determinant in the following topics:

• Double Integrals in Polar Coordinates

• Triple Integrals in Spherical Coordinates

• Linear Approximation

  • Linear approximations for vector functions of a vector variable are obtained in the same way, with the derivative at a point replaced by the Jacobian matrix.

• Curve Sketching

  • Determine the $x$- and $y$-intercepts of the curve.
  • Determine the symmetry of the curve.
  • Determine any bounds on the values of $x$ and $y$.
  • If the curve passes through the origin then determine the tangent lines there.
  • Determine the asymptotes of the curve.

• Calculating Limits Using the Limit Laws

  • Limits of functions can often be determined using simple laws, such as L'Hôpital's rule and squeeze theorem.
  • Limits of functions can often be determined using simple laws.
  • Application (or repeated application) of the rule often converts an indeterminate form to a determinate form, allowing easy evaluation of the limit.
  • The differentiation of the numerator and denominator often simplifies the quotient and/or converts it to a determinate form, allowing the limit to be evaluated more easily.

• Concavity and the Second Derivative Test

  • The second derivative test is a criterion for determining whether a given critical point is a local maximum or a local minimum.
  • In calculus, the second derivative test is a criterion for determining whether a given critical point of a real function of one variable is a local maximum or a local minimum using the value of the second derivative at the point.
  • In the latter case, Taylor's Theorem may be used to determine the behavior of $f$ near $x$ using higher derivatives.
  • A related but distinct use of second derivatives is to determine whether a function is concave up or concave down at a point.

• Maximum and Minimum Values

  • The second partial derivative test is a method used to determine whether a critical point is a local minimum, maximum, or saddle point.
  • The second partial derivative test is a method in multivariable calculus used to determine whether a critical point $(a,b, \cdots )$ of a function $f(x,y, \cdots )$ is a local minimum, maximum, or saddle point.
  • Apply the second partial derivative test to determine whether a critical point is a local minimum, maximum, or saddle point

• Equations of Lines and Planes

  • A line is a vector which connects two points on a plane and the direction and magnitude of a line determine the plane on which it lies.
  • Every line lies in a plane which is determined by both the direction and slope of the line.

• The Substitution Rule

  • Substitution can be used to determine antiderivatives if one chooses a relation between $x$ and $u$, determines the corresponding relation between $dx$ and $du$ by differentiating, and performs the substitutions.
  • An antiderivative for the substituted function can hopefully be determined; the original substitution between $u$ and $x$ is then undone.
  • Similar to our first example above, we can determine the following antiderivative with this method:

• Double Integrals Over General Regions

  • $x$-axis: If the domain $D$ is normal with respect to the $x$-axis, and $f:D \to R$ is a continuous function, then $\alpha(x)$  and $\beta(x)$ (defined on the interval $[a, b]$) are the two functions that determine $D$.
  • $y$-axis: If $D$ is normal with respect to the $y$-axis and $f:D \to R$ is a continuous function, then $\alpha(y)$ and $\beta(y)$ (defined on the interval $[a, b]$) are the two functions that determine $D$.
  • To apply the formulae, you must first find the functions that determine $D$ and the intervals over which these are defined.