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Conservative Vector Fields, Multivariable Calculus

(Unit 6 Lecture 2) We look at conservative vector fields (or gradient vector fields), which can be viewed as the gradient of a scalar-valued potential function. This lecture introduces conservative vector fields, a special category of vector fields in mathematics. These vector fields are also known as gradient vector fields because they are the gradients of scalar-valued functions (which we then call potential functions). We discuss the properties that define conservative vector fields, demonstrate how to identify them using partial derivatives, and introduce the concept of the curl in three-dimensional spaces. Note: for this discussion of conservative/gradient vector fields, we assume we are working over a "simply connected" domain (a domain with no holes). You may assume for this discussion that we are considering only vector fields defined on the entire space. (We will revisit this distinction later.) A conservative vector field is the gradient of a scalar-valued "potential function" 𝑓: 𝐹⃗ =∇𝑓. The scalar potential function's level sets, called equipotentials, represent points where the function 𝑓 has constant value. We look at a test to check if a vector field is conservative, and introduce the notion of curl. In ℝ^2, a vector field is conservative if the partial derivative of its first component with respect to 𝑦 equals the partial derivative of its second component with respect to 𝑥: 𝑃_𝑦=𝑄_𝑥. In ℝ^3, the conservativeness of a vector field is determined using the curl, which must be zero for the field to be conservative. Note: for this discussion of conservative/gradient vector fields, we assume we are working over a "simply connected" domain (a domain with no holes). You may assume for this discussion that we are considering only vector fields defined on the entire space. (We will revisit this distinction later.) #calculus #multivariablecalculus #mathematics #iitjammathematics #calculus3 #vectorfields #gradient #mathematicslecture #Conservativevectorfields #mathtutorial