The TangentPlane(f, var1, var2) command computes the plane tangent to the surface f at the point specified by var1 and var2. The surface can be entered as a Vector, a Vector-valued procedure, a scalar expression, or a scalar-valued procedure. If f is entered as a procedure, then it must have precisely two parameters.

If f is a scalar expression or a scalar-valued procedure, then the current coordinate system must be three-dimensional; i.e., cartesian (in three dimensions), spherical, or cylindrical.

If f is a scalar expression, then in order for Maple to parameterize the surface, the second and third arguments must be equations in which the left-hand sides are two distinct coordinate variable names from the current coordinate system. The surface is then parameterized using these two variables names. For example, if the coordinate system is , and the second and third arguments are  and , respectively, then f is parameterized as the Vector <, , f>. The tangent plane will be computed at the point , .

If f is a scalar-valued procedure, the second and third arguments must be equations in which the left-hand sides are two distinct coordinate variable names from the current coordinate system. These two variable names will then be plugged into f, producing a scalar expression, and the previous paragraph applies.

If the two parameter names of f coincide with precisely two of the three coordinate variable names from the current coordinate system, then there is no longer a need for the second and third arguments to specify those names, and thus they can be omitted. For example, if f is the procedure (, ) ->  +  and the current coordinate system is , then the calling sequence TangentPlane(f, 2, 3) is equivalent to the calling sequence TangentPlane(f, , ).

If f is a procedure, the information in var1 will always be substituted into the first parameter of f, and likewise, the information in var2 will always be substituted into the second parameter.

If f is a Vector, var1 and var2 must specify the names of the two parameters of the surface.

If f is a Vector-valued procedure, the names can be omitted from the second and third arguments. If they are not omitted, then these supplied names will take precedence and they will be plugged into f to produce a Vector.

If the input curve is a Vector or a scalar, the output is a position Vector. If the input curve is a procedure, the output is a procedure that evaluates to a position Vector.

If the current coordinate system is not indexed by its coordinate variable names, then the default names will be used, should they be requested during computation. For example, if the current coordinate system is cartesian (in three dimensions), then if required, the default names x, y and z will be used as the coordinate variable names.