The file DifferentiationApplications.mth defines functions that automate some of the most common applications of differentiation.  These applications include parametric, polar and implicit derivatives, curvature, centers-of-curvature, tangent lines, perpendicular lines, osculating circles, tangent planes, and surface normals.  There are separate groups of functions for curves defined in rectangular (Cartesian) coordinates either explicitly, parametrically, or implicitly, and for curves explicitly defined in polar coordinates.  The function definitions in the file are automatically loaded when any of its functions are first used.


The following functions are for curves explicitly defined in rectangular coordinates by an abscissa and an ordinate, where the ordinate is the value of the expression y that depends on the abscissa variable x:


CURVATURE(y, x) simplifies to the curvature of expression y in terms of variable x.  If you want the curvature at a particular value of x, then substitute that value in the result or take the limit of the result as x approaches that value.


CENTER_OF_CURVATURE(y, x) simplifies to the Cartesian coordinate vector of the center of curvature of expression y in terms of its variable x.


TANGENT(y, x, x0) simplifies to the line tangent to expression y(x) at x=x0.  The result will be linear in variable x.  TANGENT(y, x, inf) simplifies to the asymptote of y(x) as x approaches infinity.


PERPENDICULAR(y, x, x0) simplifies to the line perpendicular to expression y at x=x0.  The result will be linear in variable x.


OSCULATING_CIRCLE(y, x, θ) simplifies to a parametric representation of the circle that osculates expression y at its variable x, using θ as the angular parameter.


The following functions are for curves parametrically defined in rectangular coordinates by a vector of the form v = [x, y], where x and y are expressions that depend on the parameter t:


PARA_DIF(v, t, n) simplifies to the nth derivative of y with respect to x in terms of parametric variable t.  n must be a nonnegative integer, and it defaults to 1.  To learn how to program such defaults, inspect PARA_DIF's definition in the file DIF_APPS.


PARA_CURVATURE(v, t) similarly simplifies to the curvature of parametric curve v in terms of parametric variable t.


PARA_CENTER_OF_CURVATURE(v, t) similarly simplifies to a Cartesian coordinate vector of the center of curvature of v in terms of parametric variable t.


PARA_TANGENT(v, t, t0, x) simplifies to an expression linear in variable x.  This result is the y coordinate of the line tangent to parametric curve v at parameter value t=t0.


PARA_PERPENDICULAR(v, t, t0, x) similarly simplifies to the line perpendicular to parametric curve v at parameter value t=t0.


PARA_OSCULATING_CIRCLE(v, t, t0, Φ) simplifies to a parametric representation of the circle that osculates parametric curve v at parameter value t=t0.  Variable Φ is the angular parameter of this circle.


The following functions are for curves explicitly defined in polar coordinates by an expression r that depends on the angle θ:


POLAR_DIF(r, θ, n) simplifies to the nth derivative of y with respect to x in terms of variable θ.  n must be a nonnegative integer, and it defaults to 1.


POLAR_CURVATURE(r, θ) simplifies to the curvature of expression r in terms of variable θ.


POLAR_CENTER_OF_CURVATURE(r, θ) simplifies to the rectangular coordinates of the center of curvature of expression r, in terms of variable θ.


POLAR_TANGENT(r, θ, θ0, x) simplifies to an expression linear in variable x.  This result is the y coordinate of the line tangent to expression r at angle θ=θ0.


POLAR_PERPENDICULAR(r, θ, θ0, x) similarly simplifies to the line perpendicular to expression r at angle θ=θ0.


POLAR_OSCULATING_CIRCLE(r, θ, θ0, Φ) simplifies to the parametric circle that osculates polar curve r at angle θ=θ0.  Variable Φ is used as the angular circle parameter.


The following functions are for curves implicitly defined in rectangular coordinates by an equation of the form u=0, where u is an expression that depends on the variables x and y.  The values of the arguments for x and y must be unassigned variables.  Do not declare y to be a function.  The results are unique only modulo u(x,y)=0 or u(x0,y0)=0, so the symbolic results may differ from results derived by other methods.  However, the numerical results from substituting a pair of numbers that satisfy u(x,y)=0 or u(x0,y0)=0 should be the same, within roundoff error:


IMP_DIF(u, x, y, n) simplifies to the nth implicit derivative of y with respect to x.  n must be a positive integer, and it defaults to 1.  In most cases the result depends on both x and y rather than x or y alone.  This is acceptable for some applications.  If not, you can sometimes use the original equation u=0 to eliminate either y or x from the derivative.  Moreover, you can always substitute a particular value of x or of y into u, then solve for the other variable.  This may require approximate mode.  Either way, the resulting numerical values of x and y can then be substituted into the derived general implicit derivative.


IMP_CURVATURE(u, x, y) simplifies to the curvature of the curve implicitly defined by u=0.  


IMP_CENTER_OF_CURVATURE(u, x, y) simplifies to the center of curvature of the curve implicitly defined by u=0.


IMP_TANGENT(u, x, y, x0, y0) simplifies to an expression linear in variable x.  The result is the y coordinate of the line tangent at x=x0 and y=y0 to the curve implicitly defined by u=0.


IMP_PERPENDICULAR(u, x, y, x0, y0) similarly simplifies to the line perpendicular at x=x0 and y=y0 to the curve implicitly defined by u=0.


IMP_OSCULATING_CIRCLE(u, x, y, x0, y0, θ) simplifies to the circle that osculates at x=x0 and y=y0 the curve implicitly defined by u=0.  The circle is defined parametrically in terms of variable θ.



The following functions implement applications of derivatives to surfaces defined implicitly by an equation of the form u=0, where u is an expression that depends on the three rectangular (Cartesian) coordinate variables comprising the vector v.  Typically v is the vector [x, y, z]:


TANGENT_PLANE(u, v, v0) simplifies to an expression that is linear in the three coordinate variables of v.  This result defines the plane tangent to the surface u=0 at v=v0:  When equated to zero, the result is a linear relationship between the variables in v, and you can solve this equation for any of its variables in terms of the others (see the Solve commandsSolve_commands).  Given any two components of v0, you can substitute them into u, and then solve for the remaining component.  Solving it may require switching to approximate mode.


NORMAL_LINE(u, v, v0, t) simplifies to a Cartesian coordinate vector that is linear in variable t.  This result parametrically defines the line perpendicular to the surface u=0 at v=v0.  v is a vector of coordinate variables, and v0 is a corresponding vector of coordinate values on the surface.


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