Is multivariable calculus hard?
Introduction to Multivariable Calculus
Multivariable Calculus extends the concepts of single variable calculus to multiple variables. When people add more variables to calculus and perceive it in higher dimensions, it helps people solve more real-life problems. This introduction will cover the important concepts and some real-life application of multivariable calculus.
First, we need to understand the concept of “vector”.
A vector can represent the magnitude and direction of a quantity, for example the velocity of a moving object. There are many useful techniques related to vectors. Since vectors can be added, subtracted, and multiplied. For vectors’ addition, it could assist physicists or engineers to perform force analysis. For vectors’ multiplication, it involves two types of multiplication methods. One is called a “dot product”. Which helps people determine the angle between two vectors and can help calculate the projection of one vector to another.
Another multiplication method, “cross product”, helps people find the normal vector. Which is a vector that’s orthogonal to both multiplied vectors. Given the concepts of vectors, we will be able to describe a line by setting an initial point and a direction vector in a parametric form. We can also describe a plane using a point on the plane and its normal vector.
Then, we can combine the idea of vectors with functions, which is to write vector-valued functions. At this step, we could apply the techniques of calculus to vector-valued functions. We can calculate the derivative and express it in a tangent vector, which can represent the rate of change for each variable. The integral can also be calculated in the same method and be used to measure the arc length of a line, the curvature is like the second derivative and can be used to measure the rate of change of a tangent vector.
It then comes to functions with multiple variables, such as f(x,y) = ax+by+c.
In this form, it is easy to find level curves and draw contour maps. Which are very important for geographers. The idea of limit also exists in this case. And we can find the limits by either direct computing for continuous function or using squeeze theorem for non-continuous cases. Partial derivatives also exist in the multivariable case, and it helps people determine how the value of a function changes in respect to a specific variable.
In 3-D, we can also find tangent planes for a point on a function, which can be helpful in doing linear approximation. The very last concept is called a gradient vector. I will not mention the specific formulas here, but I will share the importance of gradient vectors. Firstly, it can help people find directional derivatives, which is the rate of change of a function in a specific direction. Secondly, it can also provide the direction of maximum rate of change of a function.
The above introduction briefly summarized the important concepts in multivariable calculus, and some real-life applications. To sum up, the techniques in multivariable calculus evolved from single variable calculus. However, they are more closely related to real-life situations, and could function as a useful tool in analyzing functions.