Curves in 2D is the slope of secant lines or rather the lim as h goes to zero...
So if you had some 3d object like I dont know a sphere a torus a foot ball or something with an easy equation how would you diferentiate we used implicit differentiation in calc 1 and I know multi var calc has multiple variables so... Should I study that, really I want to know how to interpret the derivative visualy.
So if you had some 3d object like I dont know a sphere a torus a foot ball or something with an easy equation how would you diferentiate we used implicit differentiation in calc 1 and I know multi var calc has multiple variables so... Should I study that, really I want to know how to interpret the derivative visualy.
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First let me get the terminology straight. A curve is a one dimensional object that can be projected into higher dimensions (think of a string). Any other multi-dimensional object would be considered a surface (a sheet of paper, or a shape like a football). That being said, differentiating either is very easy computationally and the geometric idea translates very well. To represent a multi-dimensional curve one needs a vector equation (essentially the same thing as a set of parametric equations). In this case you would simply differentiate each component just as you would in one dimension, since each component is a function of only one variable (usually time). The geometric interpretation is the same as in one dimension as well, a tangent line, moreover a line that is going in the exact same direction at the point of differentiation as the curve (you can be tangent and not parallel in 3D).
For a surface, an equation of two variables, we need a new type of differentiation called Partial Differentiation. In partial differentiation, we hold one of the variables constant while differentiating in the other. Geometrically this can be thought of as simply differentiating in one dimension (inside a plane) that isn't the xy-plane. Doing this is both variables (if in 3D) and arranging them appropriately a tangent plane is produced (a plane that contains every line tangent to the point of differentiation).
I've pulled some links from KhanAcademy. The first two deal with partial derivatives and their relation to surfaces, while the last three deal with vector valued functions (curves in space) and their derivatives and geometric implications. All of these are under the Calculus section on the main KhanAcademy page.
http://www.khanacademy.org/math/calculus…
http://www.khanacademy.org/math/calculus…
http://www.khanacademy.org/math/calculus…
http://www.khanacademy.org/math/calculus…
http://www.khanacademy.org/math/calculus…
For a surface, an equation of two variables, we need a new type of differentiation called Partial Differentiation. In partial differentiation, we hold one of the variables constant while differentiating in the other. Geometrically this can be thought of as simply differentiating in one dimension (inside a plane) that isn't the xy-plane. Doing this is both variables (if in 3D) and arranging them appropriately a tangent plane is produced (a plane that contains every line tangent to the point of differentiation).
I've pulled some links from KhanAcademy. The first two deal with partial derivatives and their relation to surfaces, while the last three deal with vector valued functions (curves in space) and their derivatives and geometric implications. All of these are under the Calculus section on the main KhanAcademy page.
http://www.khanacademy.org/math/calculus…
http://www.khanacademy.org/math/calculus…
http://www.khanacademy.org/math/calculus…
http://www.khanacademy.org/math/calculus…
http://www.khanacademy.org/math/calculus…