Math Problem Statement
Solution
This task involves finding the partial derivatives of the scalar field with respect to , , and , and then constructing the gradient vector field . Let's proceed step by step:
Given function:
Steps:
-
Partial derivative with respect to : Use the product rule .
-
Partial derivative with respect to : Take into account the term and its dependence on .
-
Partial derivative with respect to : Focus on the term , which only involves directly.
-
Construct : Combine the partial derivatives into a vector field:
I will now compute these derivatives explicitly. sympy import symbols, diff, sqrt, ln
Define variables
x, y, z = symbols('x y z')
Define the scalar field M
M = (3*x2 + y3/2 - z**4/2) * ln(x + sqrt(y))
Compute partial derivatives
partial_x = diff(M, x) partial_y = diff(M, y) partial_z = diff(M, z)
Gradient vector field (∇M)
gradient_M = (partial_x, partial_y, partial_z) partial_x, partial_y, partial_z, gradient_M### Results: The partial derivatives of are as follows:
-
Partial derivative with respect to :
-
Partial derivative with respect to :
-
Partial derivative with respect to :
Gradient vector field ():
= \left( 6x \ln(x + \sqrt{y}) + \frac{3x^2 + \frac{y^3}{2} - \frac{z^4}{2}}{x + \sqrt{y}}, \ \frac{3y^2}{2} \ln(x + \sqrt{y}) + \frac{3x^2 + \frac{y^3}{2} - \frac{z^4}{2}}{2\sqrt{y}(x + \sqrt{y})}, \ -2z^3 \ln(x + \sqrt{y}) \right).$$ This gradient forms a **vector field**, which represents the rate and direction of change of the scalar field $$M(x, y, z)$$ in space. --- Would you like a detailed explanation of any specific derivative? Or help visualizing this gradient field? ### Related Questions: 1. How do we compute directional derivatives using the gradient field? 2. What does the scalar field $$M(x, y, z)$$ physically represent in this context? 3. How can we interpret the components of $$\nabla M$$ geometrically? 4. How would the gradient behave if $$y \to 0$$? 5. Can the vector field $$\nabla M$$ be classified as conservative? ### Tip: The gradient vector always points in the direction of the steepest ascent of the scalar field $$M(x, y, z)$$.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Partial derivatives
Gradient vector
Scalar fields
Formulas
Product rule: d(fg)/dx = f'g + fg'
Gradient: ∇M = (∂M/∂x, ∂M/∂y, ∂M/∂z)
Theorems
-
Suitable Grade Level
University-level Mathematics or Advanced High School Calculus
Related Recommendation
Calculate the Gradient Vector Field of f(x, y, z) = x^3 * y * e^(y/z)
Gradient of the Function f(x, y, z) = x^2 + y^2 - 3z^2 + z ln(x) at (1, 5, 3)
Gradient of the Function f(x, y, z) = x² + y² - 2z² + zln(x)
Find the Magnitude of the Gradient for Scalar Field u = x^2/2 + y^3/3 at (1,3)
Mixed Partial Derivative of log(x² + y² + z²) and e^(x² + y² + z²)