The Laplacian operator, denoted as ∇², can be expressed in two forms: as a scalar operator and as a vector operator. The confusion arises from the notation where the unit vectors i, j, k are retained in the vector form, specifically in the expression ∇² = (i∂²/∂x² + j∂²/∂y² + k∂²/∂z²). This distinction is crucial for understanding how the Laplacian operates in different contexts, particularly in vector calculus. The sources referenced, including Wikipedia articles on the Laplacian and Vector Laplacian, provide further clarification on this topic.
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laramman2 said:When two vectors are dotted, the result is a scalar. But why here http://www.cobalt.chem.ucalgary.ca/ziegler/educmat/chm386/rudiment/mathbas/vectors.htm , the del-squared still maintains its unit vectors i, j, k? Isn't it this way ∇2 = (∂2/∂x2 + ∂2/∂y2 + ∂2/∂z2) and not (i∂2/∂x2 + j∂2/∂y2 + k∂2/∂z2)? Thank you! :)